财务管理货币时间价值Chapter3

1、8 - 1nFuture valuenPresent valuenAnnuitiesnRates of returnnAmortizationCHAPTER 3Time Value of Money8 - 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 2.8 - 3Time line for a $100 lump

2、 sum due at the end of Year 2.100012 Yeari%8 - 4Time line for an ordinary annuity of $100 for 3 years.1001001000123i%8 - 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%-508 - 6 SELF-TEST QUESTION I P.958 - 7Which would you prefer - $10,

3、000 today or $10,000 in 5 years? You already recognize that there is !The Interest Rate8 - 8Types of InterestInterest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Interest paid (earned) on only the original amount, or principal, borrowed (lent).8 - 9Sim

4、ple InterestSI = SI:Simple InterestP0:Deposit today (t=0)i:Interest Rate per Periodn:Number of Time PeriodsP0(1 + i x n).8 - 10Simple Interest Examplen Assume that you deposit $1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?8 -

5、 11is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate.Simple Interest (FV)n What is the () of the deposit?FVn = PV0 x(1 + i x n) 8 - 12Compound Interest (FV) Whats the FV of an initial $100 after 3 years if i = 10%?FV = ?012310%

6、The process of going from todays values, or present values, to future Values is called compounding.1008 - 13After 1 year:FV1= PV + INT1 = PV + PV (i)= PV(1 + i)= $100(1.10)= $110.00.After 2 years:FV2= PV(1 + i)2= $100(1.10)2= $121.00.8 - 14After 3 years:FV3= PV(1 + i)3= $100(1.10)3= $133.10.In gener

7、al,FVn= PV(1 + i)n.8 - 15 Compound Interest (FV) nFuture Value Interest Factor for i and n (FVIFi,n) :The future value of $1 left on deposit for n periods at a rate of i percent per period.nGeneral Future Value Formula:FVn =P0 x (F/P,i,n) or FVn = P0(FVIFi,n 复利终值系数复利终值系数) FVIFi,n is found on Table A

8、-3 at the end of the book.8 - 16Three Ways to Find FVsnSolve the equation with a regular calculator.nUse a financial calculator.nUse a spreadsheet.8 - 17Financial calculators solve this equation:There are 4 variables. If 3 are known, the calculator will solve for the 4th. FVPVinn 1.Financial Calcula

9、tor Solution8 - 18310-100 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.INPUTSOUTPUT8 - 19Julie Miller wants to know how large her deposit of today will become at a compound annual interest rate of 10% fo

10、r . Story Problem Example 0 1 2 3 4 10%8 - 2010%Whats the PV of $100 due in 3 years if i = 10%?Finding PVs is discounting, and its the reverse of compounding.1000123PV = ?8 - 21Solve FVn = PV(1 + i )n for PV: PV = FV1+i = FV11+innnn PV = $10011.10 = $100 0.7513 = $75.13. 38 - 22Assume that you need

11、in Lets examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 7%PV1Present Value Single Deposit (Graphic)8 - 23 =Present Value Single Deposit (Formula) 0 1 7%8 - 24nPresent Value Interest Factorfor i and n (PVIFi,n): The present value o

12、f $1 due n periods in the future discounted at i percent per period.nGeneral Present Value Formula: PV0=FVn x (P/F,i,n) or PV0 = FVn (PVIFi,n 复利现值系数复利现值系数) PVIFi,n is found on Table A-1 at the end of the book.General Present Value Formula8 - 25Julie Miller wants to know how large of a deposit to mak

13、e so that the money will grow to in at a discount rate of 10%.Story Problem Example 0 1 2 3 4 10%8 - 26Power of compound interestnSuppose one of your more frugal ancestors had invested $5 for you at a 6 percent interest rate 200 years ago.How much would you have today?8 - 27Financial Calculator Solu

14、tion3 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.INPUTSOUTPUT8 - 28Finding the Time to Double20%2012?-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 = 3.8.8

15、- 29 20 -1 0 2NI/YR PV PMTFV 3.8INPUTSOUTPUTFinancial Calculator8 - 30 SELF-TEST QUESTION II P.103 SELF-TEST QUESTION II P.1048 - 31Types of Annuities(年金）年金）: Payments or receipts occur at the end of each period.: Payments or receipts occur at the beginning of each period. represents a series of equ

16、al payments (or receipts) occurring over a specified number of equidistant periods.8 - 32Parts of an Annuity0 1 2 3 $100 $100 $100(Ordinary Annuity) ofPeriod 1 ofPeriod 2Today Cash Flows Each 1 Period Apart ofPeriod 38 - 33Parts of an Annuity0 1 2 3$100 $100 $100(Annuity Due) ofPeriod 1 ofPeriod 2To

17、day Cash Flows Each 1 Period Apart ofPeriod 38 - 34Ordinary AnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity DueWhats the difference between an ordinary annuity and an annuity due?PVFV8 - 35Examples of Annuitiesn Student Loan Paymentsn Car Loan Paymentsn Insurance Premiumsn Mortgage Paymentsn Retirement

18、 Savings8 - 36 =Overview of an Ordinary Annuity - FVA R R R R0 1 2 nR = Periodic Cash FlowCash flows occur at the end of the periodi%. . .8 - 37nFuture Value Interest Factor for an Annuity (FVIFAi,n):The future value interest factor for an annuity of n periodsn compounded at i percent.nGeneral Formu

19、la:= x (F/A,i,n) n or = (i,n 年金终值系年金终值系数数)n - Overview of an Ordinary Annuity - FVA8 - 38Whats the FV of a 3-year ordinary annuity of $100 at 10%?100100100012310% 110 121FV= 3318 - 39 =Overview of anOrdinary Annuity - PVA R R R R0 1 2 R = Periodic Cash Flowi%. . .Cash flows occur at the end of the p

20、eriod8 - 40 = Example of anOrdinary Annuity - PVA$1,000 $1,000 $1,0000 1 2 47%$934.58$873.44 $816.30Cash flows occur at the end of the period8 - 41n Present Value Interest Factor for an Annuityn (PVIFAi,n):The present value interest factorn for an annuity of n periods discounted at i n percent.n Gen

21、eral Formula:= x (P/A,i,n) or = (i,n 年金现值系数年金现值系数) - Example of anOrdinary Annuity - PVA8 - 42Whats the PV of this ordinary annuity?100100100012310%90.9182.64 75.13248.69 = PV8 - 43Find the FV and PV if theannuity were an annuity due.100100012310%1008 - 44 =Overview View of anAnnuity Due - FVAD R R

22、R R R0 1 2 3 ni%. . .Cash flows occur at the beginning of the period8 - 45 =Example of anAnnuity Due - FVAD$1,000 $1,000 $1,000 $1,0700 1 2 47%$1,225$1,145Cash flows occur at the beginning of the period8 - 46Example of anAnnuity Due - FVADFormula shown as follows:= x (F/A,i,n) x (1+i)or = (i,n ) x (

23、1+i)8 - 47Example of anAnnuity Due - FVADFormula shown as follows:FVAn=A x (F/A,i,n+1)-Aor FVAn= PMT (FVIFAi,n+1 -1)8 - 48 =Overview of anAnnuity Due - PVAD R R R R0 1 2 nR: Periodic Cash Flowi%. . .Cash flows occur at the beginning of the period8 - 49 =Example of anAnnuity Due - PVAD$1,000.00 $1,00

24、0 $1,0000 1 2 4= 7%$ 934.58$ 873.44Cash flows occur at the beginning of the period8 - 50Example of anAnnuity Due - PVADPVADn=A x (P/A,i,n) x(1+i)or PVADn= PMT (PVIFAi,n)x (1+i) 8 - 51Example of anAnnuity Due - PVAD PVADn=A x (P/A,i,n-1) or PVADn= PMT( PVIFAi,n-1)+ PMT 8 - 52Excel Function for Annuit

25、ies 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 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)8 - 53 SELF-TEST QUESTION II P.107 SELF-TE

26、ST QUESTION II P.1098 - 54Julie Miller will receive the set of cash flows below. What is the at a discount rate of . Mixed Flows Example 0 1 2 3 4 8 - 55 SELF-TEST QUESTION II P.1138 - 561. Solve a “” by discounting each back to t=0.2. Solve a “” by firstbreaking problem into groups of annuity strea

27、ms and any single cash flow groups. Then discount each back to t=0. How to Solve?8 - 57 “Piece-At-A-Time” 0 1 2 3 4 10%8 - 58“Group-At-A-Time” 0 1 2 3 4 10%$600(PVIFA10%,2) = $600(1.736) = $1,041.60$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57$100 (PVIF10%,5) = $100 (0.621) = $62.108 -

28、59“Group-At-A-Time” 0 1 2 3 4 equals 0 1 2 0 1 2 3 4 58 - 60What is the PV of this uneven cashflow stream?010013002300310%-504 90.91247.93225.39-34.15530.08 = PV8 - 61Spreadsheet SolutionExcel Formula in cell A3: =NPV(10%,B2:E2)ABCDE1012342100300300-503530.098 - 62perpetuityAnnuities go on indefinit

29、ely, or perpetuallyPV=PMT/i8 - 63 Example of an perpetuitynIf $100 is received each year forever and the interest rate is 8 percent, what is the present value of this perpetuity?PVA=8 - 641. Read problem thoroughly2. Create a time line3. Put cash flows and arrows on time line4. Determine if it is a

30、PV or FV problem5. Determine if solution involves a single CF, annuity stream(s), or mixed flow6. Solve the problemSteps to Solve Time Value of Money Problems8 - 65Will 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

31、morefrequent than once a year-for example, semiannually, quarterly,or daily-interest is earned on interestmore often.8 - 66012310%01235%456134.01100133.101230100Annually: FV3 = $100(1.10)3 = $133.10.Semiannually: FV6 = $100(1.05)6 = $134.01.8 - 67We will deal with 3 different rates:iNom = nominal, o

32、r stated, or quoted, rate per year.iPer= periodic rate.EAR= EFF% = .effective annualrate8 - 68When is each rate used?n iNOMwritten into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.n iPERUsed in calculations and shown on time lines. If m = 1, iNOM = iPER =

33、EAR.n EAR Used to compare returns on investments with different payments per year. Used in calculations when annuity payments dont match compounding periods.8 - 69niNom is stated in contracts. Periods per year (m) must also be given.nExamples:n8%; Quarterlyn8%, Daily interest (365 days)8 - 70nPeriod

34、ic rate = 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.nExamples:8% quarterly: iPer = 8%/4 = 2%.8% daily (365): iPer = 8%/365 = 0.021918%.8 - 71nEffective Annual Rate (EAR = EFF%):The annual rate of intere

35、st actually being earned, as opposed to the quoted rate. Also called the“equivalent annual rate.”.EFF% = - 1(1 + )iNommm8 - 72n An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.n Suppos

36、e you could borrow using either a credit card that charges 1 percent per month or a bank loan with a 12 percent quoted nominal interest rate that is compounded quarterly. Which should you choose?n Banks say “interest paid daily.” Same as compounded daily.8 - 73How do we find EFF% for a nominal rate

37、of 10%, compounded semiannually?Or use a financial calculator.EFF% = - 1(1 + )iNommm = - 1.0(1 + )0.1022 = (1.05)2 - 1.0 = 0.1025 = 10.25%.8 - 74FV of $100 after 3 years under 10% semiannual compounding? Quarterly?= $100(1.05)6 = $134.01.FV3Q = $100(1.025)12 = $134.49.FV = PV 1 .+ imnNommn FV = $100

38、 1 + 0.1023S2x3 8 - 75Can 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.8 - 76When is each rate used?iNom:Written into contracts, quoted by banks and brokers. Not used in ca

39、lculations or shownon time lines.8 - 77iPer:Used in calculations, shown on time lines.If iNom has annual compounding,then iPer = iNom/1 = iNom.8 - 78(Used for calculations if and only ifdealing with annuities where payments dont match interest compounding periods.)EAR = EFF%:Used to compare returns

40、on investments with different payments per year.8 - 79Whats 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 1001008 - 80nPayments occur annually, but compounding occurs each 6 months.nSo we cant use nor

41、mal annuity valuation techniques.8 - 811st Method: Compound Each CF01100235%456100100.00110.25121.55331.80FVA3= $100(1.05)4 + $100(1.05)2 + $100= $331.80. 8 - 82Could you find the FV with afinancial calculator?Yes, by following these steps:a. Find the EAR for the quoted rate:2nd Method: Treat as an

42、AnnuityEAR = (1 + ) - 1 = 10.25%. 0.10228 - 833 10.25 0 -100 INPUTS OUTPUT NI/YRPVFVPMT331.80b. Use EAR = 10.25% as the annual rate in your calculator:8 - 84Whats the PV of this stream?010015%2310010090.7082.2774.62247.598 - 85AmortizationConstruct an amortization schedulefor a $1,000, 10% annual ra

43、te loanwith 3 equal payments.8 - 86Step 1: Find the required payments.PMTPMTPMT012310%-1,0003 10 -1000 0 INPUTS OUTPUT NI/YRPVFVPMT402.118 - 87Step 2: Find interest 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 = $302.11.8 - 88Step 4: Find ending balance after Year 1.End bal = Beg