1、2-1nFuture valuenPresent valuenRates of returnnAmortizationChapter 2Time Value of Money2-2Time 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 2.2-3Time line for a$100 lump sum due at the end
2、of Year 2.100012 Yeari%2-4Time line for an ordinary annuity of$100 for 3 years.1001001000123i%2-5Time line for uneven CFs:-$50 at t=0 and$100,$75,and$50 at the end of Years 1 through 3.100 50 750123i%-502-6Whats the FV of an initial$100 after 3 years if i=10%?FV=?012310%Finding FVs(moving to the rig
3、hton a time line)is called compounding.1002-7After 1 year:FV1=PV+INT1=PV+PV(i)=PV(1+i)=$100(1.10)=$110.00.After 2 years:FV2=FV1(1+i)=PV(1+i)(1+i)=PV(1+i)2=$100(1.10)2=$121.00.2-8After 3 years:FV3=FV2(1+i)=PV(1+i)2(1+i)=PV(1+i)3=$100(1.10)3=$133.10.In general,FVn=PV(1+i)n.2-9Three Ways to Find FVsnSo
4、lve the equation with a regular calculator.nUse a financial calculator.nUse a spreadsheet.2-10Financial calculator:HP17BIInAdjust display contrast:hold down CLR and push+or-.nChoose algebra mode:Hold down orange key(i.e.,the shift key),hit MODES(the shifted DSP key),and select ALG.nSet number of dec
5、imal places to display:Hit DSP key,select FIX,then input desired decimal places(e.g.,3).2-11HP17BII(Continued)nSet decimal mode:Hit DSP key,select the“.”instead of the“,”.Note:many non-US countries reverse the US use of decimals and commas when writing a number.2-12HP17BII:Set Time Value Parametersn
6、Hit EXIT until you get the menu starting with FIN.Select FIN.nSelect TVM.nSelect OTHER.nSelect P/YR.Input 1(for 1 payment per year).nSelect END(for cash flows occuring at the end of the year.)2-13Financial calculators solve this equation:There are 4 variables.If 3 are known,the calculator will solve
7、 for the 4th.0ni1PVnFVFinancial Calculator Solution2-14310-100 0NI/YR PV PMTFV 133.10Heres the setup to find FV:Clearing automatically sets everything to 0,but for safety enter PMT=0.Set:P/YR=1,END.INPUTSOUTPUT2-15Spreadsheet SolutionnUse the FV function:see spreadsheet in Ch 02 Mini Case.xls.l=FV(R
8、ate,Nper,Pmt,PV)l=FV(0.10,3,0,-100)=133.102-1610%Whats the PV of$100 due in 3 years if i=10%?Finding PVs is discounting,and its the reverse of compounding.1000123PV=?2-17Solve FVn=PV(1+i)n for PV:PV =FV1+i=FV11+innnn PV=$10011.10 =$100 0.7513 =$75.13.32-18Financial Calculator Solution3 10 0100N I/YR
9、 PV PMTFV -75.13Either PV or FV must be negative.HerePV=-75.13.Put in$75.13 today,take out$100 after 3 years.INPUTSOUTPUT2-19Spreadsheet SolutionnUse the PV function:see spreadsheet.l=PV(Rate,Nper,Pmt,FV)l=PV(0.10,3,0,100)=-75.132-20Finding the Time to Double20%2012?-1 FV=PV(1+i)n$2=$1(1+0.20)n (1.2
10、)n=$2/$1=2nLN(1.2)=LN(2)n=LN(2)/LN(1.2)n=0.693/0.182=3.8.2-21 20 -1 0 2NI/YR PV PMTFV 3.8INPUTSOUTPUTFinancial Calculator2-22Spreadsheet SolutionnUse the NPER function:see spreadsheet.l=NPER(Rate,Pmt,PV,FV)l=NPER(0.10,0,-1,2)=3.82-23Finding the Interest Rate?%20123-1 FV=PV(1+i)n$2=$1(1+i)3 (2)(1/3)=
11、(1+i)1.2599=(1+i)i=0.2599=25.99%.2-243 -1 0 2NI/YR PV PMTFV25.99INPUTSOUTPUTFinancial Calculator2-25Spreadsheet SolutionnUse the RATE function:l=RATE(Nper,Pmt,PV,FV)l=RATE(3,0,-1,2)=0.25992-26Ordinary AnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity DueWhats the difference between an ordinary annuity an
12、d an annuity due?PVFV2-27Whats the FV of a 3-year ordinary annuity of$100 at 10%?100100100012310%110 121FV=3312-28FV Annuity FormulanThe future value of an annuity with n periods and an interest rate of i can be found with the following formula:.33110.1000.101)0(1i1i)(1PMT3n2-29Financial calculators
13、 solve this equation:There are 5 variables.If 4 are known,the calculator will solve for the 5th.0i1ni)(1PMTni1PVnFVFinancial Calculator Formula for Annuities2-303 10 0 -100 331.00NI/YRPVPMTFVFinancial Calculator SolutionHave payments but no lump sum PV,so enter 0 for present value.INPUTSOUTPUT2-31Sp
14、readsheet SolutionnUse the FV function:see spreadsheet.l=FV(Rate,Nper,Pmt,Pv)l=FV(0.10,3,-100,0)=331.002-32Whats the PV of this ordinary annuity?100100100012310%90.9182.64 75.13248.69 =PV2-33PV Annuity FormulanThe present value of an annuity with n periods and an interest rate of i can be found with
15、 the following formula:69.24810.1000.10)0(111-ii)(111-PMT3n2-34Have payments but no lump sum FV,so enter 0 for future value.3 10 100 0NI/YRPVPMTFV-248.69INPUTSOUTPUTFinancial Calculator Solution2-35Spreadsheet SolutionnUse the PV function:see spreadsheet.l=PV(Rate,Nper,Pmt,Fv)l=PV(0.10,3,100,0)=-248
16、.692-36Find the FV and PV if theannuity were an annuity due.100100012310%1002-37PV and FV of Annuity Due vs.Ordinary AnnuitynPV of annuity due:l=(PV of ordinary annuity)(1+i)l=(248.69)(1+0.10)=273.56nFV of annuity due:l=(FV of ordinary annuity)(1+i)l=(331.00)(1+0.10)=364.12-38310 100 0 -273.55 NI/YR
17、PVPMTFVSwitch from“End”to“Begin”.Then enter variables to find PVA3=$273.55.Then enter PV=0 and press FV to findFV=$364.10.INPUTSOUTPUT2-39Excel Function for Annuities DueChange the formula to:=PV(10%,3,-100,0,1)The fourth term,0,tells the function there are no other cash flows.The fifth term tells t
18、he function that it is an annuity due.A similar function gives the future value of an annuity due:=FV(10%,3,-100,0,1)2-40What is the PV of this uneven cashflow stream?010013002300310%-504 90.91247.93225.39-34.15530.08 =PV2-41nInput in“CFLO”register:CF0 =0CF1 =100CF2 =300CF3 =300CF4 =-50nEnter I=10%,
19、then press NPV button to get NPV=530.09.(Here NPV=PV.)2-42Spreadsheet SolutionExcel Formula in cell A3:=NPV(10%,B2:E2)ABCDE1012342100300300-503530.092-43Nominal rate(iNom)n Stated in contracts,and quoted by banks and brokers.n Not used in calculations or shown on time linesn Periods per year(m)must
20、be given.n Examples:l8%;Quarterlyl8%,Daily interest(365 days)2-44Periodic rate(iPer)n iPer=iNom/m,where m is number of compounding periods per year.m=4 for quarterly,12 for monthly,and 360 or 365 for daily compounding.n Used in calculations,shown on time lines.n Examples:l8%quarterly:iPer=8%/4=2%.l8
21、%daily(365):iPer=8%/365=0.021918%.2-45Will the FV of a lump sum be larger or smaller if we compound more often,holding the stated I%constant?Why?LARGER!If compounding is morefrequent than once a year-for example,semiannually,quarterly,or daily-interest is earned on interestmore often.2-46FV Formula
22、with Different Compounding Periods(e.g.,$100 at a 12%nominal rate with semiannual compounding for 5 years)=$100(1.06)10 =$179.08.FV =PV 1 .+imnNommn FV =$100 1+0.1225S2x5 2-47FV 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)
23、10=$179.08.FV(Quarterly)=$100(1.03)20=$180.61.FV(Monthly)=$100(1.01)60=$181.67.FV(Daily)=$100(1+(0.12/365)(5x365)=$182.19.2-48Effective Annual Rate(EAR=EFF%)n The EAR is the annual rate which causes PV to grow to the same FV as under multi-period compounding Example:Invest$1 for one year at 12%,semi
24、annual:FV=PV(1+iNom/m)m FV=$1(1.06)2 =1.1236.EFF%=12.36%,because$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.2-49nAn investment with monthly payments is different from one with quarterly payments.Must put on
25、 EFF%basis to compare rates of return.Use EFF%only for comparisons.nBanks say“interest paid daily.”Same as compounded daily.2-50How do we find EFF%for a nominal rate of 12%,compounded semiannually?EFF%=-1(1+)iNommm =-1.0(1+)0.1222 =(1.06)2-1.0 =0.1236=12.36%.2-51Finding EFF with HP17BIInGo to menu s
26、tarting TVM.nSelect ICNV(for int.rate conversion).nSelect PER(for periodic compounding).nEnter nominal rate and select NOM%.nEnter number of periods per year and select P.nSelect EFF%,which returns effective rate.2-52EAR(or EFF%)for a Nominal Rate of of 12%EARAnnual=12%.EARQ=(1+0.12/4)4-1=12.55%.EAR
27、M=(1+0.12/12)12-1=12.68%.EARD(365)=(1+0.12/365)365-1=12.75%.2-53Can the effective rate 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.2-54When is each rate used?iNom:Written into contracts,quoted by b
28、anks and brokers.Not used in calculations or shownon time lines.2-55iPer:Used in calculations,shown on time lines.If iNom has annual compounding,then iPer =iNom/1=iNom.2-56(Used for calculations if and only ifdealing with annuities where payments dont match interest compounding periods.)EAR=EFF%:Use
29、d to compare returns on investments with different payments per year.2-57AmortizationConstruct an amortization schedulefor a$1,000,10%annual rate loanwith 3 equal payments.2-58Step 1:Find the required payments.PMTPMTPMT012310%-1,0003 10 -1000 0 INPUTS OUTPUT NI/YRPVFVPMT402.112-59Step 2:Find interes
30、t charge for Year 1.INTt=Beg balt(i)INT1=$1,000(0.10)=$100.Step 3:Find repayment of principal in Year 1.Repmt=PMT-INT =$402.11-$100 =$302.11.2-60Step 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
31、.2-61Interest declines.Tax implications.BEGPRINENDYRBALPMTINTPMTBAL1$1,000$402$100$302$6982698402703323663366402373660TOT1,206.34206.341,0002-62$0123402.11Interest302.11Level payments.Interest declines because outstanding balance declines.Lender earns10%on loan outstanding,which is falling.Principal
32、 Payments2-63nAmortization tables are widely used-for home mortgages,auto loans,business loans,retirement plans,and so on.They are very important!nFinancial calculators(and spreadsheets)are great for setting up amortization tables.2-64On January 1 you deposit$100 in an account that pays a nominal in
33、terest rate of 11.33463%,with daily compounding(365 days).How much will you have on October 1,or after 9 months(273 days)?(Days given.)2-65iPer=11.33463%/365=0.031054%per day.FV=?0122730.031054%-100Note:%in calculator,decimal in equation.FV=$100 1.00031054 =$100 1.08846=$108.85.2732732-66273-100 0 1
34、08.85INPUTSOUTPUTNI/YRPVFVPMTiPer=iNom/m=11.33463/365=0.031054%per day.Enter i in one step.Leave data in calculator.2-67Whats the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%,compounded semiannually?01100235%456 6-mos.periods 1001002-68nPayments occur annu
35、ally,but compounding occurs each 6 months.nSo we cant use normal annuity valuation techniques.2-691st Method:Compound Each CF01100235%456100100.00110.25121.55331.80FVA3=$100(1.05)4+$100(1.05)2+$100=$331.80.2-70Could you find the FV with afinancial calculator?Yes,by following these steps:a.Find the E
36、AR for the quoted rate:2nd Method:Treat as an AnnuityEAR=(1+)-1=10.25%.0.10222-713 10.25 0 -100 INPUTS OUTPUT NI/YRPVFVPMT331.80b.Use EAR=10.25%as the annual rate in your calculator:2-72Whats the PV of this stream?010015%2310010090.7082.2774.62247.592-73You are offered a note which pays$1,000 in 15
37、months(or 456 days)for$850.You have$850 in a bank which pays a 6.76649%nominal rate,with 365 daily compounding,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.Should you buy it?2-743 Ways to Solve:1.Greatest f
38、uture wealth:FV2.Greatest wealth today:PV3.Highest rate of return:Highest EFF%iPer=0.018538%per day.1,0000365456 days-8502-751.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$924.97.2-76456-
39、850 0 924.97INPUTSOUTPUTNI/YRPVFVPMTCalculator Solution to FV:iPer=iNom/m=6.76649%/365=0.018538%per day.Enter iPer in one step.2-772.Greatest Present WealthFind PV of note,and comparewith its$850 cost:PV=$1,000/(1.00018538)456=$918.95.2-78 456 .018538 0 1000 -918.95INPUTSOUTPUTNI/YRPVFVPMT6.76649/36
40、5=PV of note is greater than its$850 cost,so buy the note.Raises your wealth.2-79Find 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 Return2-80 456-850 0 1000 0.035646%per day INPUTSOUTPUTNI/YRPVFVPMTConvert%to decimal:Decimal=0.035646/100=0.00035646.EAR=EFF%=(1.00035646)365-1 =13.89%.2-81Using interest conversion:P/YR=365NOM%=0.035646(365)=13.01 EFF%=13.89Since 13.89%7.0%opportunity cost,buy the note.
