1、Chapter 3Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan1Time Value of Money“A dollar today is worth than a dollar tomorrow.”Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and
2、 Wachowicz.Slides prepared by Wu Xiaolan2Chapter ObjectiveslDistinguish between simple and compound interest.lCalculate the present value and future value of a single amount for both one period and multiple periods.lCalculate the present value and future value of multiple cash flows.lCalculate the p
3、resent value and future value of annuities.lCompare nominal interest rates(NIR)and effective annual interest rates(EAR).lDetermine the amortization schedule.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan3When youre
4、in your 20s,youre young.Who thinks about retiring at this age?.Youre just beginning to make some money.Perhaps there are college loans to be paid.If youre in your 30s,youve probably started a family.Youre scraping enough together to buy your first home and make the mortgage payments.Now in your 40s,
5、you may be facing demands on your earnings.You need a larger home.The kids are taking dance and piano lessons.they need braces.You may have some financial responsibility for aging parents.By your 50s,you have children in college.You may be experiencing late-in-career job changes or setbacks.Suddenly
6、 youre 60,and for all the best reasons in the world you didnt save along the way.For so long it seemed so far away.Retirement is now upon you,and you arent ready financially.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xi
7、aolan4Cash-Flow Time LineTime periods0Cash flow-in(现金流入)Cash flow-out(现金流出)12345Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan5Time Value TerminologylFuture value(FV)is the amount an investment is worth after one or
8、 more periods.lPresent value(PV)is the current value of future cash flows of an investment.01234PVFVCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan6Time Value TerminologylThe number of time periods between the presen
9、t value and the future value is represented by n or“t”.lThe rate of interest for discounting or compounding is called i or“r”.lAll time value questions involve包括 four values:PV,FV,i and n.Given three of them,it is always possible to calculate the fourth.Copyright 2001 Prentice-Hall,Inc.Fundamentals
10、of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan7Interest Rate TerminologylSimple interest refers to interest earned only on the original capital investment amount.lCompound interest复利 refers to interest earned on both the initial capital investment and on the in
11、terest reinvested from prior period.“Worlds eighth wonder”the power of compounding.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan8Future ValuesExample-Simple InterestInterest earned at a rate of 6%for five years on
12、a principal balance of$100.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan9Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.Interest Earned Per Year=100
13、 x .06 =$6Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan10Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2 3 4 5Interest EarnedVa
14、lue100Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan11Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2 3 4 5Interest Earned 6Valu
15、e100106Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan12Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2 3 4 5Interest Earned 6 6V
16、alue100106112Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan13Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2 3 4 5Interest Earne
17、d 6 6 6Value100106112118Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan14Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2 3 4 5Int
18、erest Earned 6 6 6 6 Value100106112118124Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan15Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Y
19、ears 1 2 3 4 5Interest Earned 6 6 6 6 6Value100106112118124130Value at the end of Year 5=$130Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan16lFor any simple interest rate,the future value of an account invested toda
20、y at the end of n periods is:FP(1+ni)lThe present value of an account is:P=F(1+ni)-1Simple Interest FormulaCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan17Future ValuesExample-Compound InterestInterest earned at a r
21、ate of 6%for five years on the previous years balance.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan18Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years balanc
22、e.Interest Earned Per Year=Prior Year Balance x .06Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan19Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years balance.T
23、odayFuture Years 1 2 3 4 5Interest EarnedValue100Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan20Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years balance.Tod
24、ayFuture Years 1 2 3 4 5Interest Earned 6.00Value100106.00Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan21Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years ba
25、lance.TodayFuture Years 1 2 3 4 5Interest Earned 6.00 6.36Value100106.00 112.36Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan22Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on
26、the previous years balance.TodayFuture Years 1 2 3 4 5Interest Earned 6.00 6.36 6.74Value100106.00 112.36 119.10Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan23Future Values?Example-Compound InterestInterest earned
27、at a rate of 6%for five years on the previous years balance.TodayFuture Years 1 2 3 4 5Interest Earned 6.00 6.36 6.74 7.15Value100106.00 112.36 119.10 126.25Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan24Future Val
28、ues?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years balance.TodayFuture Years 1 2 3 4 5Interest Earned 6.00 6.36 6.74 7.15 7.57Value100106.00 112.36 119.10 126.25 133.82Value at the end of Year 5=$133.82Copyright 2001 Prentice-Hall,Inc.Fundamentals of Fin
29、ancial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan25Future Value of a Lump SumlThe accumulated value of this investment at the end of five years can be split into two components:original principal$100interest earned$33.82lUsing simple interest,the total interest earned w
30、ould only have been$30.The other$3.82 is from compounding.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan26Future Value of a Lump SumlIn general,the future value,FVn of an account invested today at i%for t periods is
31、:F=P(1+i)nlThe expression(1+i)n is the future value interest factor(FVIFi,n).Refer to Table I.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan27Future Values with CompoundingInterest RatesCopyright 2001 Prentice-Hall,
32、Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan28The Rule of 72lThe Rule of 72 is a handy rule of thumb that states:If you earn r%per year,your money will double in about 72/r%years.lFor example,if you invest at 6%,your money will double in abou
33、t 12 years.lThis rule is only an approximate近似的 rule.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan29Present Value of a Lump Sum汇总lThe future value,PVn of an account invested today at i%for t periods is:P=F(1+i)-nlT
34、he expression(1+i)-n is the present value interest factor(PVIFi,n).Refer to Table II.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan30Present Value of$1 for Different Periods and RatesPresentvalueof$1($)Time(years)r=
35、0%r=5%r=10%r=15%r=20%1 2 3 4 5 6 7 8 9 101.00.90.80.70.60.50.40.30.20.10Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan31PV of Multiple多样的 Cash FlowslPVs can be added together to evaluate multiple cash flows.PVCrCr11
36、2211()().Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan32PV of Multiple Cash FlowsExampleYour auto dealer gives you the choice to pay$15,500 cash now,or make three payments:$8,000 now and$4,000 at the end of the fol
37、lowing two years.If your cost of money is 8%,which do you prefer?$15,133.06 PVTotal36.429,370.703,38,000.0021)08.1(000,42)08.1(000,41payment ImmediatePVPVCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan33Types of Annu
38、itieslAn Annuity(年金)represents a series of equal payments(or receipts)occurring over a specified number of equidistant periods.Ordinary Annuity（普通年金）:Payments or receipts occur at the end of each period.Annuity Due（先付年金）:Payments or receipts occur at the beginning of each period.A perpetuity（永续年金）is
39、 an annuity in which the cash flows continue forever.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan34Examples of Annuitiesl Student Loan Paymentsl Car Loan Paymentsl Insurance Premiumsl Mortgage Paymentsl Retirement
40、 SavingsCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan35Future Value of An Annuity01234AAAAA（1+i）0A（1+i）1A（1+i）2A（1+i）3FCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wac
41、howicz.Slides prepared by Wu Xiaolan36 F=F1+F2+F3+Fn =A+A(1+i)1+A(1+i)2+A(1+i)n-1 (1+i)n1 =A iFuture Value of An AnnuitylThe compounding term is called the future value interest factor for annuities(FVIFAi,n).Refer to Table III.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11
42、/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan37Present Value of An Annuity 01234AAAAA/（1+i）1A/（1+i）2A/（1+i）3A/（1+i）4PCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan38Present Value of An Annuity P=A(1+i)
43、-1+A(1+i)-2+A(1+i)-n 1(1+i)-n =A ilThe discounting term is called the present value interest factor for annuities(PVIFAi,n).Refer to Table IV.Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan39Example-AnnuityYou are pu
44、rchasing a car.You are scheduled to make 3 annual installments of$4,000 per year.Given a rate of interest of 10%,what is the price you are paying for the car(i.e.what is the PV)?PVPV4 000947 41110110 1103,$9,.(.)Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne
45、and Wachowicz.Slides prepared by Wu Xiaolan40Hint on Annuity ValuationlThe present value of an ordinary annuity can be viewed as occurring at the beginning of the first cash flow period.lThe future value of an ordinary annuity can be viewed as occurring at the end of the last cash flow period.Copyri
46、ght 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan41PerpetuitieslThe future value of a perpetuity cannot be calculated as the cash flows are infinite.lThe present value of a perpetuity is calculated as follows:P=A(1+i)-1+A(1+
47、i)-2+A(1+i)-n 1(1+i)-n =A i n,1/(1+i)n0 P=A/I Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan42 Julie Miller will receive the set of cash flows below.What is the Present Value at a discount rate of 10%?0 1 2 3 4$600$
48、600$400$400$100PV0Mixed Flows ExampleCopyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan431.Solve a“piece-at-a-time”by discounting each piece back to t=0.2.Solve a“group-at-a-time”by first breaking problem into groups of
49、 annuity streams and any single cash flow group.Then discount each group back to t=0.How to Solve?Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan44 0 1 2 3 4 5$600$600$400$400$10010%$545.45$495.87$300.53$273.21$62.09
50、$1677.15=PV0“Piece-At-A-Time”Copyright 2001 Prentice-Hall,Inc.Fundamentals of Financial Management,11/e by Van Horne and Wachowicz.Slides prepared by Wu Xiaolan45 0 1 2 3 4 5$600$600$400$400$10010%$1,041.60$573.57$62.10$1,677.27=PV0“Group-At-A-Time”(#1)$600(PVIFA10%,2)=$600(1.736)=$1,041.60$400(PVIF
