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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/YRPMTPVFV3.82002-16-12Future value of an annuitynAn annuity is a series of equal payments made at fixed intervals for a specified number of periodsnOrdinary(deferred)annuity:nIf the payments occur at the end of each periodnAnnuity due:nPayments are made at the beginnin
9、g of each period6-13FV of Ordinary Annuities5%0100110021003105110.25FVA3=315.256-14FV of Annuities Due0110021003105115.76FVA3=331.015%100110.256-15What is the difference between an ordinary annuity and an annuity due?Ordinary Annuity(普通年金普通年金)PMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity Due(期初年金期初年金)6-16Th
10、e 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 invests the accumulated savings($1,095)in an online stock account.The stock account has an expected annual return of 12%.How
11、much money will she have when she is 65 years old?查表求解6-17PV of Ordinary Annuities011002395.2486.38PVA3=272.325%10090.701006-18What is the PV of this uneven cash flow stream?010013002300310%-504 90.91247.93225.39-34.15530.08 =PV6-19PerpetuitiesnWhen annuities go on perpetuallynIt is called perpetuit
12、y6-20Proof6-21Solving 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.)6-22Will the FV of a lump sum be larger or smaller if compounded more often,holding the stated
13、 I%constant?nLARGER,as the more frequently compounding occurs,interest is earned on interest more often.Annually(每年複利每年複利):FV3=$100(1.10)3=$133.10012310%100133.10Semiannually(半年複利半年複利):FV6=$100(1.05)6=$134.0101235%456134.0112301006-23Classifications of interest ratesnNominal rate(iNOM)also called th
14、e Annual Percentage Rate(or APR),or quoted or state rate.(名目或報價利率)nThe is the rate the banks,credit card companies,and so on tell you that they are charging on the loansnIt is also the rate banks say they are paying on deposits6-24Classifications of interest ratesnEffective(or equivalent)annual rate
15、(EAR=EFF%)(有效年利率)the annual rate of interest actually being earned,taking into account compounding.n一般通稱的利率為名目利率,常以年表示,即利息1年複息1次。但由於現實複利期間並非皆為1年,因此不同複利期間所形成的有效利率也會有所不同。n有效利率是在特定付息條件下的真正利率水準,有助於比較當複利期間不同時,所產生的實際報酬為何?6-25Classifications of interest ratesnYou can convert any nominal rate to an equivale
16、nt annual rateWhere“m”is the number of compounding periods per year.n亦即,可視為能夠和每年複利一次的利率產生相同未來值的利率。6-26nFor Example,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%annu
17、al return,compounded semiannually.6-27Another examplenMany banks charge 1.5%per month,and,in their advertising,they state that the Annual Percentage Rate is 1.5%x12=18%nHowever,the effective annual rate is 19.6%6-28When is each rate used?niNOMwritten into contracts,quoted by banks and brokers.Not us
18、ed in calculations or shown on time lines.nEARUsed to compare returns on investments with different payments per year.Used in calculations when annuity payments dont match compounding periods.6-29Why is it important to consider effective rates of return?nAn investment with monthly payments is differ
19、ent 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%EARQUARTERLY10.38%EARMONTHLY10.47%EARDAILY(365)10.52%6-30Can the effective rate ever be equ
20、al 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.6-31What is the FV of$100 after 3 years under 10%semiannual compounding?Quarterly compounding?6-32Whats the FV of a 3-year$100 annuity,if the quoted interest rate i
21、s 10%,compounded semiannually?nPayments occur annually,but compounding occurs every 6 months.nCannot use normal annuity valuation techniques.01100235%4510010061236-33Method 1:Compound each cash flow110.25121.55331.80FV3=$100(1.05)4+$100(1.05)2+$100FV3=$331.8001100235%4510061231006-34Method 2:Financi
22、al calculatornFind the EAR and treat as an annuity.nEAR=(1+0.10/2)2 1=10.25%.INPUTSOUTPUTNI/YRPMTPVFV310.25-100331.8006-35Find 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-
23、247.596-36Loan amortization(貸款攤銷)nAmortization tables are widely used for home mortgages,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 wit
24、h 3 equal payments.6-37Step 1:Find the required annual paymentnAll input information is already given,just remember that the FV=0 because the reason for amortizing the loan and making payments is to retire the loan.INPUTSOUTPUTNI/YRPMTPVFV310402.110-10006-38Step 2:Find the interest paid in Year 1nTh
25、e borrower will owe interest upon the initial balance at the end of the first year.Interest 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)=$1006-39Step 3:Find the principal repaid in Year 1nIf a payment of$402.11
26、was made at the end of the first year and$100 was paid toward interest,the remaining value must represent the amount of principal repaid.PRIN=PMT INT=$402.11-$100=$302.116-40Step 4:Find the ending balance after Year 1nTo find the balance at the end of the period,subtract the amount paid toward princ
27、ipal from the beginning balance.END BAL=BEG BAL PRIN=$1,000-$302.11=$697.896-41Constructing 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$6982698402703323663366402373660TOTAL1,206.34206.341,000-6-42Illustrating an amortized payment:Where does the money go?nConstant payments.nDeclining interest payments.nDeclining balance.$0123402.11Interest302.11Principal Payments
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