第三章貨幣的時(shí)間價(jià)值chapter3-Timevalueofmoney

1、3-1Do you have these puzzles?nBuy a mobile by lump-sum payment or by installment ?nArrange savings for a future expenditure?nWhat kind of loans to apply for?nBuy or sell a bond you are holding?n 2021-10-16Ch3 Time Value of Money3-2CHAPTER 3Time Value of MoneynCompounding and Discounting of Single Su

2、msnAnnuitiesnTypes of Interest Rates3-32021-10-16Ch3 Time Value of MoneyCH3 The Time Value of MoneyTo find the answer, youll have to know3-42021-10-16Ch3 Time Value of Money3.1 Compounding and Discounting Single Sums3-52021-10-16Ch3 Time Value of MoneyWe know that receiving $1 today is worth more th

3、an $1 in the future. This is due to opportunity costs.The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.TodayFuture3-62021-10-16Ch3 Time Value of MoneyIf we can measure this opportunity cost, we can:nTranslate $1 today into its e

4、quivalent in the future (compounding).nTranslate $1 in the future into its equivalent today (discounting).?TodayFutureToday?Future3-72021-10-16Ch3 Time Value of MoneyTime 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 p

5、eriod (year, month, etc.) or the beginning of the second period.CF0CF1CF3CF20123i%3-82021-10-16Ch3 Time Value of MoneyDrawing time lines:1）$100 lump sum due in 2 years;100012i%$100 lump sum due in 2 years3-92021-10-16Ch3 Time Value of MoneyDrawing time lines:2) Uneven cash flow stream; CF0 = -$50, C

6、F1 = $100, CF2 = $75, and CF3 = $50 100 50 750123i%-50Uneven cash flow stream3-102021-10-16Ch3 Time Value of MoneyCompounding:nTo find the Future Value (FV) of a cash flow, we suppose we earn interest on principal as well as on interest accumulated each term. Today?Future3-112021-10-16Ch3 Time Value

7、 of MoneyExampleWhat is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%? FV can be solved by using the arithmetic, or Table A-3 (pp.A-6)FV = ?012310%1003-122021-10-16Ch3 Time Value of MoneySolving for FV:The arithmetic methodnAfter 1 year:nFV1 = PV ( 1 + i ) = $100 (1.10) = $11

8、0.00nAfter 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 )n3-132021-10-16Ch3 Time Value of MoneySolving for FV:The table-checking methodFVn = PV *FVIF(i, n) FV3= $100* FVIF(10%,3) = $100*

9、1.331 =$133.103-142021-10-16Ch3 Time Value of MoneyDiscounting:nFinding the Present Value (PV) of a cash flow or series of cash flows when compound interest is applied (the reverse of compounding).?TodayFuture3-152021-10-16Ch3 Time Value of MoneyPV = ?100ExampleWhat is the present value (PV) of $100

10、 due in 3 years, if I/YR = 10%? 012310%3-162021-10-16Ch3 Time Value of MoneySolving for PV:The arithmetic methodnSolve the general FV equation for PV:nPV = FVn / ( 1 + i )nnPV = FV3 / ( 1 + i )3 = $100 / ( 1.10 )3 = $75.133-172021-10-16Ch3 Time Value of MoneySolving for PV:The table-checking methodn

11、Solve the general FV equation for PV:nPV = FVn * PVIF(i,n)Ref.: Table A-1 (PP.A-2)nPV = FV3 * PVIF(10% ,3) = $100 *0.7513 = $75.133-182021-10-16Ch3 Time Value of MoneynSolves the general FV or PV equation for N.nSame as previous problems, but now solving for N. Eg. Pp. 104Other application 1: Solvin

12、g for NWith I, PV, FV known to you 3-192021-10-16Ch3 Time Value of MoneynSolves the general FV or PV equation for i.nSame as previous problems, but now solving for N. Eg. Pp. 103Other application 2: Solving for IWith n, PV, FV known to you 3-202021-10-16Ch3 Time Value of MoneyA magic 72 rulenWhen th

13、e compound interest rate is less than 20%, the years you need to double your todays wealth is 72/the interest rate number.nConversely ,if you want to double your wealth in X years, you have to ensure your investment earn 72/X% annually!3-212021-10-16Ch3 Time Value of MoneyThe Power of Compound Inter

14、estn compounding the 8th wonder!nThe power of compound interest exceeds the nuclear bomb!3-222021-10-16Ch3 Time Value of MoneyStory of an ancient AtheniannThis Athenian pocketed all the money except a single Drachma, which he invested in Athenian government bonds paying 3 percent compounded annually

15、. nHe didnt live long enough to see the results, but after 2,000 years that Drachmae wound up being worth more than all the assets on the Earth!3-232021-10-16Ch3 Time Value of MoneynTo make the “nuclear bomb” work, investing early, or, a long enough investment period is necessary! 3-242021-10-16Ch3

16、Time Value of MoneynSuppose the merchant invests only 3 years, he will have only 1*FVIF(3%,3)=1.0927 Drachmae3-252021-10-16Ch3 Time Value of Moneyna sequence of equal cash flows, occurring at fixed intervals for a specified number of periods. 012343.2 Annuities3-262021-10-16Ch3 Time Value of Moneyna

17、 bonds semi-annual coupon interest payments over the life of the bond.nRepaying bank loans: a stream of equal repayments.Examples of Annuities:3-272021-10-16Ch3 Time Value of MoneyTypes of annuities:Ordinary (deferred) annuityAnnuity duePerpetuities3-282021-10-16Ch3 Time Value of MoneyDifference bet

18、ween an ordinary annuity and an annuity dueOrdinary AnnuityPMTPMTPMT0123i%PMTPMT0123i%PMTAnnuity Due3-292021-10-16Ch3 Time Value of Money3.2.1 Solving for FV and PV of ordinary annuities 3-302021-10-16Ch3 Time Value of MoneynFVAn= A* FVIFA(i,n)Eg. What is the FV of 3-year ordinary annuity of $100 at

19、 10%?nFVA3= $100* FVIFA(10%, 3) = $100* 3.310 =$331.0Solving for FV of ordinary annuity:3-312021-10-16Ch3 Time Value of MoneyHow about this? Solve for FV0 1 2 3 4100 100 100FVA3=100*FVIFA(10%,3) =100*3.310=331.0010%0 1 2 3 4100 100 1003-322021-10-16Ch3 Time Value of MoneynPVA n = A* PVIFA(i,n)Eg. Wh

20、at is the PV of 3-year ordinary annuity of $100 at 10%?nPVAn= $100* PVIFA(10%, 3) = $100* 2.487 =$248.70exercises:nP.127 3-6Solving for PV of ordinary annuity:3-332021-10-16Ch3 Time Value of MoneyHow about this? Solve for PV0 1 2 3 4100 100 100100*PVIFA(10%,3)*PVIF(10%,1) =100*2.487*0.9091=226.0910%

21、3-342021-10-16Ch3 Time Value of Money3.2.2 Solving for FV and PV of annuities due3-352021-10-16Ch3 Time Value of MoneySolving for FV of annuity due:nIt can be calculated by adjusting the FV equation for ordinary annuity:FVAn(annuity due)= A* FVIFA(i,n+1)-13-362021-10-16Ch3 Time Value of MoneyEg. Wha

22、t is the FV of 3-year annuity due of $100 at 10%?FVA3(annuity due)= 100* FVIFA(i,n+1)-1 =100*FVIFA(10%, 4)-1 =100*(4.641-1) =364.1Exercises：nP.127 3-73-372021-10-16Ch3 Time Value of MoneySolving for PV of annuity due:nIt can be calculated by adjusting the PV equation for ordinary annuity:PVAn(annuit

23、y due)= A* PVIFA(i,n-1)+13-382021-10-16Ch3 Time Value of MoneynEg. What is the PV of 3-year annuity due of $100 at 10%?nPVA3 (annuity due) = A* PVIFA(i,n-1)+1 = 100* PVIFA(10%,2)+1 =100* (1.736+1)= $273.60Exercises：nP.127 What is the present value of the cash flow in 3-7?3-392021-10-16Ch3 Time Value

24、 of Money3.2.3 PerpetuitiesnPerpetuity is a stream of equal payments expected to continue forever. 0 1 2 3 4A A A A.3-402021-10-16Ch3 Time Value of MoneynCan you decide the FV of a perpetuity?nPV of a perpetuity: PV (perpetuity) = A / IEg. John wants to establish a fund to aid the handicapped $10000

25、 per year.Suppose the annual investment return of the fund is expected to be 8% for ever, how much money should he invest today?PV (perpetuity) = A / I=10000/0.08=125,000.003-412021-10-16Ch3 Time Value of MoneyUneven cash flow stream010013002300310%-5043-422021-10-16Ch3 Time Value of MoneyWhat is th

26、e PV of this uneven cash flow stream?010013002300310%-504 90.91247.93225.39 -34.15530.08 = PV3-432021-10-16Ch3 Time Value of MoneyExercises:nP.131 3-393-442021-10-16Ch3 Time Value of MoneyAgain: power of compoundingwith the case of annuities 1. A 20-year-old student wants to start saving for retirem

27、ent. he plans to save $3 a day. Every day, he puts $3 in his drawer. At the end of the year, he 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 he have when he is 60 years old?3-452021-10-16Ch3 Time Value

28、 of MoneynIf he begins saving today, and sticks to her plan, he will have $839,963.55 when he is 60.FVA40= 1095* FVIFA(12%,40)=839963.553-462021-10-16Ch3 Time Value of Money2. If the student begins saving when he is 40 years old, he will have FVA20= 1095* FVIFA(12%,20)=$78896.94 at age 60. This is $

29、761,066.61 less than if starting at age 20.3-472021-10-16Ch3 Time Value of Money3. How much must the 40-year old deposit annually to catch the 20-year old?nthe final goal : $ 839,963.55 nPmt period: 203-482021-10-16Ch3 Time Value of MoneynFVA20= A* FVIFA(12%,20)=839963.55Solve for A:A=839963.55/ FVI

30、FA(12%,20)=$11657.74The student has to save $31.94 daily to achieve the goal.10 times of the 20 years old!3-492021-10-16Ch3 Time Value of MoneynLesson: It pays to start saving early.3-502021-10-16Ch3 Time Value of MoneynCompounding frequencies also matter!3-512021-10-16Ch3 Time Value of MoneyWill th

31、e 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%100133.10Semiannually: FV6 = $100(1.05)6 = $134.0101235%456134.01

32、12301003-522021-10-16Ch3 Time Value of Money3.3 Types of interest ratesnNominal ratenPeriodic ratenEffective annual rateRefer to p.1183-532021-10-16Ch3 Time Value of Money1) Nominal rate (iNOM) also called the quoted or stated rate. An annual rate that ignores compounding effects.niNOM is stated in

33、contracts. The number of compounding periods must also be givene.g. 8% , Quarterly or 8% , Daily.niNOM can be compared with one another only if the instruments have the same compounding period per year.niNOM is never shown on a time line; neither is it used as an input of compounding or discounting

34、calculations.3-542021-10-16Ch3 Time Value of MoneyExamplenForeign Currency Time Deposit Interest Rates (% p.a.) of HSBCCurrency 1 month 3 months Australian Dollar (AUD) 5.0900 5.2000British Pound(GBP) 1.3750 1.7500Canadian Dollar (CAD) 1.1250 1.31253-552021-10-16Ch3 Time Value of Money招商銀行招商銀行儲(chǔ)蓄存款基準(zhǔn)

35、利儲(chǔ)蓄存款基準(zhǔn)利 率率 利率單位為利率單位為 % / 年年 （2011.10）存期人民幣活期0.5整存整取三個(gè)月3.1整存整取一年3.5整存整取二年4.43-562021-10-16Ch3 Time Value of Money2) Periodic rate (iPER) the rate of interest charged each compounding/payment period, e.g. monthly or quarterly.niPER = iNOM / m, where: m is the number of compounding periods per year.E

36、g: m = 4 for quarterly and m = 12 for monthly compounding.3-572021-10-16Ch3 Time Value of Money3) Effective (or equivalent) annual rate (EAR = EFF%) the annual rate of interest actually being earned, taking into account compounding.nEFF% for 10% semiannual investmentEFF% = ( 1 + iNOM / m )m - 1= ( 1

37、 + 0.10 / 2 )2 1 = 10.25%(iNOM / mthe periodic rateMthe number of compounding periods per year)3-582021-10-16Ch3 Time Value of MoneynAn investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually.3-592021-10-16Ch3

38、Time Value of MoneynEAR is not used in calculations.nIt is used to compare the effective cost or rate of return when loans/investmens payment periods differ 3-602021-10-16Ch3 Time Value of MoneyExample:Effective rates of bank loan quotation 10%, annually/quarterly/monthly/daily are:EARANNUAL10.00%EA

39、RQUARTERLY10.38%EARMONTHLY10.47%EARDAILY (365)10.52%3-612021-10-16Ch3 Time Value of Moneyn招商銀行三個(gè)月整存整?。?.1%），自動(dòng)轉(zhuǎn)存，一年的實(shí)際利率是多少？n與一年期整存整取（3.5%）相比如何呢？EAR=3.14%3-622021-10-16Ch3 Time Value of MoneyCan the effective rate ever be equal to the nominal rate?nYes, but only if annual compounding is used, i.e.,

40、if m = 1.nIf m 1, EFF% will always be greater than the nominal rate.3-632021-10-16Ch3 Time Value of MoneyCredit cardnDo you have a credit card?nDo you know how is the interest charged?3-642021-10-16Ch3 Time Value of Moneyn建行、工行等大部分銀行:透支利息按月計(jì)收復(fù)利n農(nóng)業(yè)銀行: 透支利息按月計(jì)收取單利 n利率：大部分銀行按日息萬分之5計(jì)算，但也有例外，如： 中銀長(zhǎng)城人民幣卡規(guī)

41、定：自透支之日起15天內(nèi)按日息萬分之五計(jì)算，超過15天按日息萬分之十計(jì)算，超過30天或透支金額超過規(guī)定限額的，按日息萬分之十五計(jì)算3-652021-10-16Ch3 Time Value of MoneynIf a credit card charges DAILY interest at 0.05%, compounded monthly. What is the EFF%? Is it expensive?EFF%= ( 1 + 1.5% )12 - 1=20%3-662021-10-16Ch3 Time Value of Moneyn招商銀行招商銀行個(gè)個(gè) 人人 貸貸 款款 參參 考考 利

42、利 率率 (自2011年7月生效) 貸款年限年利率6月-1年 (含1年) 6.1%3-672021-10-16Ch3 Time Value of MoneyAgain:When is each rate used?niNOMwritten into contracts, quoted by banks and brokers. Not 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. 3-682021-10-16Ch3 Time Value of MoneyWhat is the FV of $100 after 3 years und