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1、,Future valuePresent valueRates of returnAmortization,Chapter 2Time Value of Money,Time lines show timing of cash flows.,CF0,CF1,CF3,CF2,0,1,2,3,i%,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.,Time line for a $100 lump sum due at the
2、 end of Year 2.,100,0,1,2 Year,i%,Time line for an ordinary annuity of $100 for 3 years.,100,100,100,0,1,2,3,i%,Time line for uneven CFs: -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3.,100,50,75,0,1,2,3,i%,-50,Whats the FV of an initial $100 after 3 years if i = 10%?,FV = ?,0,
3、1,2,3,10%,Finding FVs (moving to the righton a time line) is called compounding.,100,After 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.,After 3 years:,FV3= FV2(1+i)=PV(1 + i)2(1+i)= PV(1+i)3= $100
4、(1.10)3= $133.10.,In general,FVn= PV(1 + i)n.,Three Ways to Find FVs,Solve the equation with a regular calculator.Use a financial calculator.Use a spreadsheet.,Financial calculator: HP17BII,Adjust display contrast: hold down CLR and push + or -.Choose algebra mode: Hold down orange key (i.e., the sh
5、ift key), hit MODES (the shifted DSP key), and select ALG.Set number of decimal places to display: Hit DSP key, select FIX, then input desired decimal places (e.g., 3).,HP17BII (Continued),Set decimal mode: Hit DSP key, select the “.” instead of the “,”. Note: many non-US countries reverse the US us
6、e of decimals and commas when writing a number.,HP17BII: Set Time Value Parameters,Hit EXIT until you get the menu starting with FIN. Select FIN.Select TVM.Select OTHER.Select P/YR. Input 1 (for 1 payment per year).Select END (for cash flows occuring at the end of the year.),Financial calculators so
7、lve this equation:,There are 4 variables. If 3 are known, the calculator will solve for the 4th.,Financial Calculator Solution,310-100 0NI/YR PV PMTFV 133.10,Heres the setup to find FV:,Clearing automatically sets everything to 0, but for safety enter PMT = 0.,Set:P/YR = 1, END.,INPUTS,OUTPUT,Spread
8、sheet Solution,Use the FV function: see spreadsheet in Ch 02 Mini Case.xls. = FV(Rate, Nper, Pmt, PV) = FV(0.10, 3, 0, -100) = 133.10,10%,Whats the PV of $100 due in 3 years if i = 10%?,Finding PVs is discounting, and its the reverse of compounding.,100,0,1,2,3,PV = ?,Solve FVn = PV(1 + i )n for PV:
9、,PV,=,$100,1,1.10,=,$100,0.7513,=,$75.13.,3,Financial Calculator Solution,3 10 0100N I/YR PV PMTFV -75.13,Either PV or FV must be negative. HerePV = -75.13. Put in $75.13 today, take out $100 after 3 years.,INPUTS,OUTPUT,Spreadsheet Solution,Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt
10、, FV) = PV(0.10, 3, 0, 100) = -75.13,Finding the Time to Double,20%,2,0,1,2,?,-1,FV= PV(1 + i)n $2= $1(1 + 0.20)n (1.2)n= $2/$1 = 2nLN(1.2)= LN(2) n= LN(2)/LN(1.2) n= 0.693/0.182 = 3.8.,20 -1 0 2NI/YR PV PMTFV 3.8,INPUTS,OUTPUT,Financial Calculator,Spreadsheet Solution,Use the NPER function: see spr
11、eadsheet. = NPER(Rate, Pmt, PV, FV) = NPER(0.10, 0, -1, 2) = 3.8,Finding the Interest Rate,?%,2,0,1,2,3,-1,FV= PV(1 + i)n $2= $1(1 + i)3 (2)(1/3)= (1 + i) 1.2599= (1 + i) i= 0.2599 = 25.99%.,3 -1 0 2NI/YR PV PMTFV25.99,INPUTS,OUTPUT,Financial Calculator,Spreadsheet Solution,Use the RATE function: =
12、RATE(Nper, Pmt, PV, FV) = RATE(3, 0, -1, 2) = 0.2599,Ordinary Annuity,PMT,PMT,PMT,0,1,2,3,i%,PMT,PMT,0,1,2,3,i%,PMT,Annuity Due,Whats the difference between an ordinary annuity and an annuity due?,PV,FV,Whats the FV of a 3-year ordinary annuity of $100 at 10%?,100,100,100,0,1,2,3,10%,110 121FV= 331,
13、FV Annuity Formula,The future value of an annuity with n periods and an interest rate of i can be found with the following formula:,Financial calculators solve this equation:,There are 5 variables. If 4 are known, the calculator will solve for the 5th.,Financial Calculator Formula for Annuities,3 10
14、 0 -100 331.00,N,I/YR,PV,PMT,FV,Financial Calculator Solution,Have payments but no lump sum PV, so enter 0 for present value.,INPUTS,OUTPUT,Spreadsheet Solution,Use the FV function: see spreadsheet. = FV(Rate, Nper, Pmt, Pv) = FV(0.10, 3, -100, 0) = 331.00,Whats the PV of this ordinary annuity?,100,
15、100,100,0,1,2,3,10%,90.91,82.64,75.13,248.69 = PV,PV Annuity Formula,The present value of an annuity with n periods and an interest rate of i can be found with the following formula:,Have payments but no lump sum FV, so enter 0 for future value.,3 10 100 0,N,I/YR,PV,PMT,FV,-248.69,INPUTS,OUTPUT,Fina
16、ncial Calculator Solution,Spreadsheet Solution,Use the PV function: see spreadsheet. = PV(Rate, Nper, Pmt, Fv) = PV(0.10, 3, 100, 0) = -248.69,Find the FV and PV if theannuity were an annuity due.,100,100,0,1,2,3,10%,100,PV and FV of Annuity Due vs. Ordinary Annuity,PV of annuity due: = (PV of ordin
17、ary annuity) (1+i) = (248.69) (1+ 0.10) = 273.56FV of annuity due:= (FV of ordinary annuity) (1+i)= (331.00) (1+ 0.10) = 364.1,310 100 0 -273.55,N,I/YR,PV,PMT,FV,Switch from “End” to “Begin”.Then enter variables to find PVA3 = $273.55.,Then enter PV = 0 and press FV to findFV = $364.10.,INPUTS,OUTPU
18、T,Excel Function for Annuities Due,Change 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 the function that it is an annuity due. A similar function gives the future value of an annuity due:=FV(10%,3,-100,0,1),What is the P
19、V of this uneven cashflow stream?,0,100,1,300,2,300,3,10%,-50,4,90.91,247.93,225.39,-34.15,530.08 = PV,Input in “CFLO” register:CF0 = 0CF1 = 100CF2 = 300CF3 = 300CF4 = -50Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.),Spreadsheet Solution,Excel Formula in cell A3: =NPV(10
20、%,B2:E2),ABCDE1012342100300300-503530.09,Nominal rate (iNom),Stated in contracts, and quoted by banks and brokers.Not used in calculations or shown on time linesPeriods per year (m) must be given.Examples:8%; Quarterly8%, Daily interest (365 days),Periodic rate (iPer ),iPer = iNom/m, where m is numb
21、er of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.Used in calculations, shown on time lines.Examples:8% quarterly: iPer = 8%/4 = 2%.8% daily (365): iPer = 8%/365 = 0.021918%.,Will the FV of a lump sum be larger or smaller if we compound mor
22、e 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.,FV Formula with Different Compounding Periods (e.g., $100 at a 12% nominal rate with semiannual compounding fo
23、r 5 years),= $100(1.06)10 = $179.08.,FV,=,PV,1 .,+,i,m,n,Nom,mn,FV,=,$100,1,+,0.12,2,5S,2x5,FV of $100 at a 12% nominal rate for 5 years with different compounding,FV(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 =
24、 $181.67.FV(Daily)= $100(1+(0.12/365)(5x365) = $182.19.,Effective Annual Rate (EAR = EFF%),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%, semiannual: FV = PV(1 + iNom/m)m FV = $1 (1.06)2 = 1.1236.EFF% = 12.3
25、6%, 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.,An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% onl
26、y for comparisons.Banks say “interest paid daily.” Same as compounded daily.,How do we find EFF% for a nominal rate of 12%, compounded semiannually?,= - 1.0,(1 + ),0.122,2,= (1.06)2 - 1.0,= 0.1236 = 12.36%.,Finding EFF with HP17BII,Go to menu starting TVM.Select ICNV (for int.rate conversion).Select
27、 PER (for periodic compounding).Enter nominal rate and select NOM%.Enter number of periods per year and select P.Select EFF%, which returns effective rate.,EAR (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 + 0.12/36
28、5)365 - 1= 12.75%.,Can the effective rate ever be equal to the nominal rate?,Yes, but only if annual compounding is used, i.e., if m = 1.If m 1, EFF% will always be greater than the nominal rate.,When is each rate used?,iNom:,Written into contracts, quoted by banks and brokers. Not used in calculati
29、ons or shownon time lines.,iPer:,Used in calculations, shown on time lines.,If iNom has annual compounding,then iPer = iNom/1 = iNom.,(Used for calculations if and only ifdealing with annuities where payments dont match interest compounding periods.),EAR = EFF%:,Used to compare returns on investment
30、s with different payments per year.,Amortization,Construct an amortization schedulefor a $1,000, 10% annual rate loanwith 3 equal payments.,Step 1: Find the required payments.,PMT,PMT,PMT,0,1,2,3,10%,-1,000,3 10 -1000 0,INPUTS,OUTPUT,N,I/YR,PV,FV,PMT,402.11,Step 2: Find interest charge for Year 1.,I
31、NTt= 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.,Step 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 ta
32、ble.,Interest declines. Tax implications.,BEGPRINENDYRBALPMTINTPMTBAL,1$1,000$402$100$302$6982698402703323663366402373660TOT1,206.34206.341,000,$,0,1,2,3,402.11,Interest,302.11,Level payments. Interest declines because outstanding balance declines. Lender earns10% on loan outstanding, which is falli
33、ng.,Principal Payments,Amortization tables are widely used-for home mortgages, auto loans, business loans, retirement plans, and so on. They are very important!Financial calculators (and spreadsheets) are great for setting up amortization tables.,On January 1 you deposit $100 in an account that pays
34、 a nominal interest 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.),iPer= 11.33463%/365= 0.031054% per day.,FV=?,0,1,2,273,0.031054%,-100,Note: % in calculator, decimal in equation.,(,),(,),FV,=,$100,1.00031054,=,$
35、100,1.08846,=,$108.85.,273,273,273-100 0 108.85,INPUTS,OUTPUT,N,I/YR,PV,FV,PMT,iPer=iNom/m=11.33463/365=0.031054% per day.,Enter i in one step.Leave data in calculator.,Whats the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?,0,1,10
36、0,2,3,5%,4,5,6 6-mos. periods,100,100,Payments occur annually, but compounding occurs each 6 months.So we cant use normal annuity valuation techniques.,1st Method: Compound Each CF,0,1,100,2,3,5%,4,5,6,100,100.00,110.25,121.55,331.80,FVA3= $100(1.05)4 + $100(1.05)2 + $100= $331.80.,Could you find th
37、e FV with afinancial calculator?,Yes, by following these steps:a. Find the EAR for the quoted rate:,2nd Method: Treat as an Annuity,EAR = (1 + ) - 1 = 10.25%.,0.102,2,3 10.25 0 -100,INPUTS,OUTPUT,N,I/YR,PV,FV,PMT,331.80,b. Use EAR = 10.25% as the annual rate in your calculator:,Whats the PV of this
38、stream?,0,100,1,5%,2,3,100,100,90.7082.2774.62247.59,You 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 compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the mone
39、y in the bank if you dont buy the note. The note is riskless.Should you buy it?,3 Ways to Solve:1. Greatest future wealth: FV2. Greatest wealth today: PV3. Highest rate of return: Highest EFF%,iPer =0.018538% per day.,1,000,0,365,456 days,-850,1. Greatest Future Wealth,Find FV of $850 left in bank f
40、or15 months and compare withnotes FV = $1,000.,FVBank=$850(1.00018538)456=$924.97 in bank.,Buy the note: $1,000 $924.97.,456-850 0 924.97,INPUTS,OUTPUT,N,I/YR,PV,FV,PMT,Calculator Solution to FV:,iPer=iNom/m=6.76649%/365=0.018538% per day.,Enter iPer in one step.,2. Greatest Present Wealth,Find PV o
41、f note, and comparewith its $850 cost:,PV=$1,000/(1.00018538)456=$918.95.,456 .018538 0 1000 -918.95,INPUTS,OUTPUT,N,I/YR,PV,FV,PMT,6.76649/365 =,PV of note is greater than its $850 cost, so buy the note. Raises your wealth.,Find the EFF% on note and compare with 7.0% bank pays, which is your opport
42、unity cost of capital:,FVn= PV(1 + i)n,$1,000 = $850(1 + i)456,Now we must solve for i.,3. Rate of Return,456-850 0 1000 0.035646% per day,INPUTS,OUTPUT,N,I/YR,PV,FV,PMT,Convert % to decimal:,Decimal = 0.035646/100 = 0.00035646.,EAR = EFF%= (1.00035646)365 - 1 = 13.89%.,Using interest conversion: P/YR=365NOM%=0.035646(365)= 13.01 EFF%=13.89Since 13.89% 7.0% opportunity cost,buy the note.,