1、6-1CHAPTER 6Time Value of MoneynFuture valuenPresent valuenAnnuitiesnRates of returnnAmortization6-2Time linesnShow the timing of cash flows.nTick marks occur at the end of periods,so Time 0 is today;Time 1 is the end of the first period(year,month,etc.)or the beginning of the second period.CF0CF1CF
2、3CF20123i%6-3Drawing time lines:$100 lump sum due in 2 years;3-year$100 ordinary annuity1001001000123i%3 year$100 ordinary annuity100012i%$100 lump sum due in 2 years6-4Drawing time lines:Uneven cash flow stream;CF0=-$50,CF1=$100,CF2=$75,and CF3=$50 100 50 750123i%-50Uneven cash flow stream6-5What i
3、s the future value(FV)of an initial$100 after 3 years,if I/YR=10%?nFinding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding.nFV can be solved by using the arithmetic,financial calculator,and spreadsheet methods.FV=?012310%1006-6Solving for FV:The
4、arithmetic methodnAfter 1 year:nFV1=PV(1+i)=$100(1.10)=$110.00nAfter 2 years:nFV2=PV(1+i)2=$100(1.10)2 =$121.00nAfter 3 years:nFV3=PV(1+i)3=$100(1.10)3 =$133.10nAfter n years(general case):nFVn=PV(1+i)n6-7Solving for FV:The calculator methodnSolves the general FV equation.nRequires 4 inputs into cal
5、culator,and will solve for the fifth.(Set to P/YR=1 and END mode.)INPUTSOUTPUTNI/YRPMTPVFV3100133.10-1006-8PV=?100What is the present value(PV)of$100 due in 3 years,if I/YR=10%?nFinding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting(the reverse
6、of compounding).nThe PV shows the value of cash flows in terms of todays purchasing power.012310%6-9Solving for PV:The arithmetic methodnSolve the general FV equation for PV:nPV=FVn/(1+i)nnPV=FV3/(1+i)3 =$100/(1.10)3 =$75.136-10Solving for PV:The calculator methodnSolves the general FV equation for
7、PV.nExactly like solving for FV,except we have different input information and are solving for a different variable.INPUTSOUTPUTNI/YRPMTPVFV3100100-75.136-11Solving for N:If sales grow at 20%per year,how long before sales double?nSolves the general FV equation for N.nSame as previous problems,but no
8、w solving for N.INPUTSOUTPUTNI/YRPMTPVFV3.82002-16-12What is the difference between an ordinary annuity and an annuity due?Ordinary AnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity Due6-13Solving for FV:3-year ordinary annuity of$100 at 10%n$100 payments occur at the end of each period,but there is no P
9、V.INPUTSOUTPUTNI/YRPMTPVFV310-10033106-14Solving for PV:3-year ordinary annuity of$100 at 10%n$100 payments still occur at the end of each period,but now there is no FV.INPUTSOUTPUTNI/YRPMTPVFV3101000-248.696-15Solving for FV:3-year annuity due of$100 at 10%nNow,$100 payments occur at the beginning
10、of each period.nSet calculator to“BEGIN”mode.INPUTSOUTPUTNI/YRPMTPVFV310-100364.1006-16Solving for PV:3 year annuity due of$100 at 10%nAgain,$100 payments occur at the beginning of each period.nSet calculator to“BEGIN”mode.INPUTSOUTPUTNI/YRPMTPVFV3101000-273.556-17What is the PV of this uneven cash
11、flow stream?010013002300310%-504 90.91247.93225.39-34.15530.08 =PV6-18Solving for PV:Uneven cash flow streamnInput cash flows in the calculators“CFLO”register:nCF0=0nCF1=100nCF2=300nCF3=300nCF4=-50nEnter I/YR=10,press NPV button to get NPV=$530.09.(Here NPV=PV.)6-19Solving for I:What interest rate w
12、ould cause$100 to grow to$125.97 in 3 years?nSolves the general FV equation for I.INPUTSOUTPUTNI/YRPMTPVFV380125.97-1006-20The Power of Compound InterestA 20-year-old student wants to start saving for retirement.She plans to save$3 a day.Every day,she puts$3 in her drawer.At the end of the year,she
13、invests the accumulated savings($1,095)in an online stock account.The stock account has an expected annual return of 12%.How much money will she have when she is 65 years old?6-21Solving for FV:Savings problemnIf she begins saving today,and sticks to her plan,she will have$1,487,261.89 when she is 6
14、5.INPUTSOUTPUTNI/YRPMTPVFV4512-10951,487,26206-22Solving for FV:Savings problem,if you wait until you are 40 years old to startnIf a 40-year-old investor begins saving today,and sticks to the plan,he or she will have$146,000.59 at age 65.This is$1.3 million less than if starting at age 20.nLesson:It
15、 pays to start saving early.INPUTSOUTPUTNI/YRPMTPVFV2512-1095146,00106-23Solving for PMT:How much must the 40-year old deposit annually to catch the 20-year old?nTo find the required annual contribution,enter the number of years until retirement and the final goal of$1,487,261.89,and solve for PMT.I
16、NPUTSOUTPUTNI/YRPMTPVFV2512-11,154.421,487,26206-24Will the FV of a lump sum be larger or smaller if compounded more often,holding the stated I%constant?nLARGER,as the more frequently compounding occurs,interest is earned on interest more often.Annually:FV3=$100(1.10)3=$133.10012310%100133.10Semiann
17、ually:FV6=$100(1.05)6=$134.0101235%456134.0112301006-25Classifications of interest ratesnNominal rate(iNOM)also called the quoted or state rate.An annual rate that ignores compounding effects.niNOM is stated in contracts.Periods must also be given,e.g.8%Quarterly or 8%Daily interest.nPeriodic rate(i
18、PER)amount of interest charged each period,e.g.monthly or quarterly.niPER=iNOM/m,where m is the number of compounding periods per year.m=4 for quarterly and m=12 for monthly compounding.6-26Classifications of interest ratesnEffective(or equivalent)annual rate(EAR=EFF%)the annual rate of interest act
19、ually being earned,taking into account compounding.nEFF%for 10%semiannual investmentEFF%=(1+iNOM/m)m-1=(1+0.10/2)2 1=10.25%nAn investor would be indifferent between an investment offering a 10.25%annual return and one offering a 10%annual return,compounded semiannually.6-27Why is it important to con
20、sider effective rates of return?nAn investment with monthly payments is different from one with quarterly payments.Must put each return on an EFF%basis to compare rates of return.Must use EFF%for comparisons.See following values of EFF%rates at various compounding levels.EARANNUAL10.00%EARQUARTERLY1
21、0.38%EARMONTHLY10.47%EARDAILY(365)10.52%6-28Can 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.6-29When is each rate used?niNOMwritten into contracts,quoted by banks and brokers.Not
22、 used in calculations or shown on time lines.niPERUsed in calculations and shown on time lines.If m=1,iNOM=iPER=EAR.nEAR Used to compare returns on investments with different payments per year.Used in calculations when annuity payments dont match compounding periods.6-30What is the FV of$100 after 3
23、 years under 10%semiannual compounding?Quarterly compounding?$134.49 (1.025)$100 FV$134.01 (1.05)$100 FV)20.10 1 ($100 FV)mi 1 (PV FV123Q63S323SnmNOMn6-31Whats the FV of a 3-year$100 annuity,if the quoted interest rate is 10%,compounded semiannually?nPayments occur annually,but compounding occurs ev
24、ery 6 months.nCannot use normal annuity valuation techniques.01100235%4510010061236-32Method 1:Compound each cash flow110.25121.55331.80FV3=$100(1.05)4+$100(1.05)2+$100FV3=$331.8001100235%4510061231006-33Method 2:Financial calculatornFind the EAR and treat as an annuity.nEAR=(1+0.10/2)2 1=10.25%.INP
25、UTSOUTPUTNI/YRPMTPVFV310.25-100331.8006-34Find the PV of this 3-year ordinary annuity.nCould solve by discounting each cash flow,or nUse the EAR and treat as an annuity to solve for PV.INPUTSOUTPUTNI/YRPMTPVFV310.251000-247.596-35Loan amortizationnAmortization tables are widely used for home mortgag
26、es,auto loans,business loans,retirement plans,etc.nFinancial calculators and spreadsheets are great for setting up amortization tables.nEXAMPLE:Construct an amortization schedule for a$1,000,10%annual rate loan with 3 equal payments.6-36Step 1:Find the required annual paymentnAll input information i
27、s already given,just remember that the FV=0 because the reason for amortizing the loan and making payments is to retire the loan.INPUTSOUTPUTNI/YRPMTPVFV310402.110-10006-37Step 2:Find the interest paid in Year 1nThe borrower will owe interest upon the initial balance at the end of the first year.Int
28、erest to be paid in the first year can be found by multiplying the beginning balance by the interest rate.INTt=Beg balt(i)INT1=$1,000(0.10)=$1006-38Step 3:Find the principal repaid in Year 1nIf a payment of$402.11 was made at the end of the first year and$100 was paid toward interest,the remaining v
29、alue must represent the amount of principal repaid.PRIN=PMT INT=$402.11-$100=$302.116-39Step 4:Find the ending balance after Year 1nTo find the balance at the end of the period,subtract the amount paid toward principal from the beginning balance.END BAL=BEG BAL PRIN=$1,000-$302.11=$697.896-40Constru
30、cting an amortization table:Repeat steps 1 4 until end of loannInterest paid declines with each payment as the balance declines.What are the tax implications of this?YearBEG BALPMTINTPRINEND BAL1$1,000$402$100$302$6982698402703323663366402373660TOTAL1,206.34206.341,000-6-41Illustrating an amortized
31、payment:Where does the money go?nConstant payments.nDeclining interest payments.nDeclining balance.$0123402.11Interest302.11Principal Payments6-42Partial amortizationnBank agrees to lend a home buyer$220,000 to buy a$250,000 home,requiring a$30,000 down payment.nThe home buyer only has$7,500 in cash
32、,so the seller agrees to take a note with the following terms:nFace value=$22,500n7.5%nominal interest ratenPayments made at the end of the year,based upon a 20-year amortization schedule.nLoan matures at the end of the 10th year.6-43Calculating annual loan paymentsnBased upon the loan information,t
33、he home buyer must make annual payments of$2,207.07 on the loan.INPUTSOUTPUTNI/YRPMTPVFV207.52207.070-225006-44Determining the balloon paymentnUsing an amortization table(spreadsheet or calculator),it can be found that at the end of the 10th year,the remaining balance on the loan will be$15,149.54.nTherefore,nBalloon payment=$15,149.54nFinal payment=$17,356.61
