1、0Chapter 3Copyright 2001 Prentice-Hall,Inc.1Time Value of Money“A dollar today is worth than a dollar tomorrow.”Copyright 2001 Prentice-Hall,Inc.2Chapter ObjectiveslDistinguish between simple and compound interest.lCalculate the present value and future value of a single amount for both one period a
2、nd multiple periods.lCalculate the present value and future value of multiple cash flows.lCalculate the present 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.3When
3、 youre 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 yo
4、ur 40s,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.S
5、uddenly 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.4Cash-Flow Time LineTime periods0Cash flow-in(现金流入)Cash flow-out(现金流出)12345Copyright 2
6、001 Prentice-Hall,Inc.5Time Value TerminologylFuture value(FV)is the amount an investment is worth after one or more periods.lPresent value(PV)is the current value of future cash flows of an investment.01234PVFVCopyright 2001 Prentice-Hall,Inc.6Time Value TerminologylThe number of time periods betwe
7、en the present 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.
8、7Interest 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 interest reinvested from prior period.“Worlds eighth wonder”the power of compounding.Co
9、pyright 2001 Prentice-Hall,Inc.8Future ValuesExample-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.Copyright 2001 Prentice-Hall,Inc.9Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.Interest
10、Earned Per Year=100 x .06 =$6Copyright 2001 Prentice-Hall,Inc.10Future 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 EarnedValue100Copyright 2001 Prentice-Hall,Inc.11Future Values?Example-Simple InterestI
11、nterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2 3 4 5Interest Earned 6Value100106Copyright 2001 Prentice-Hall,Inc.12Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a principal balance of$100.TodayFuture Years 1 2
12、3 4 5Interest Earned 6 6Value100106112Copyright 2001 Prentice-Hall,Inc.13Future 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 6 6Value100106112118Copyright 2001 Prentice-Hall,Inc.14Future Values?
13、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 6 6 6 Value100106112118124Copyright 2001 Prentice-Hall,Inc.15Future Values?Example-Simple InterestInterest earned at a rate of 6%for five years on a princip
14、al balance of$100.TodayFuture Years 1 2 3 4 5Interest Earned 6 6 6 6 6Value100106112118124130Value at the end of Year 5=$130Copyright 2001 Prentice-Hall,Inc.16lFor any simple interest rate,the future value of an account invested today at the end of n periods is:FP(1+ni)lThe present value of an accou
15、nt is:P=F(1+ni)-1Simple Interest FormulaCopyright 2001 Prentice-Hall,Inc.17Future ValuesExample-Compound InterestInterest earned at a rate of 6%for five years on the previous years balance.Copyright 2001 Prentice-Hall,Inc.18Future Values?Example-Compound InterestInterest earned at a rate of 6%for fi
16、ve years on the previous years balance.Interest Earned Per Year=Prior Year Balance x .06Copyright 2001 Prentice-Hall,Inc.19Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years balance.TodayFuture Years 1 2 3 4 5Interest EarnedValue100Copyright 20
17、01 Prentice-Hall,Inc.20Future Values?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 6.00Value100106.00Copyright 2001 Prentice-Hall,Inc.21Future Values?Example-Compound InterestInterest earned at a rate o
18、f 6%for five years on the previous years balance.TodayFuture Years 1 2 3 4 5Interest Earned 6.00 6.36Value100106.00 112.36Copyright 2001 Prentice-Hall,Inc.22Future Values?Example-Compound InterestInterest earned at a rate of 6%for five years on the previous years balance.TodayFuture Years 1 2 3 4 5I
19、nterest Earned 6.00 6.36 6.74Value100106.00 112.36 119.10Copyright 2001 Prentice-Hall,Inc.23Future Values?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 7.15Value100106.00 112.36 119.10 12
20、6.25Copyright 2001 Prentice-Hall,Inc.24Future Values?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=$133.82Cop
21、yright 2001 Prentice-Hall,Inc.25Future 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 would only have been$30.The other$3.82 is from com
22、pounding.Copyright 2001 Prentice-Hall,Inc.26Future Value of a Lump SumlIn general,the future value,FVn of an account invested today at i%for t periods is: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.27Future Values with
23、 CompoundingInterest RatesCopyright 2001 Prentice-Hall,Inc.28The 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 about 12 years.lThis rule is only an approximat
24、e近似的 rule.Copyright 2001 Prentice-Hall,Inc.29Present 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)-nlThe expression(1+i)-n is the present value interest factor(PVIFi,n).Refer to Table II.Copyright 2001 Prentice-Hall,Inc.30Present Value of$1 fo
25、r Different Periods and RatesPresentvalueof$1($)Time(years)r=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.31PV of Multiple多样的 Cash FlowslPVs can be added together to evaluate multiple cash flows.PVCrCr112211()().Copyright 2001 Prentice-Hall
26、,Inc.32PV 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 following 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(000,41pa
27、yment ImmediatePVPVCopyright 2001 Prentice-Hall,Inc.33Types of AnnuitieslAn 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
28、receipts occur at the beginning of each period.A perpetuity（永续年金）is an annuity in which the cash flows continue forever.Copyright 2001 Prentice-Hall,Inc.34Examples of Annuitiesl Student Loan Paymentsl Car Loan Paymentsl Insurance Premiumsl Mortgage Paymentsl Retirement SavingsCopyright 2001 Prentice
29、-Hall,Inc.35Future Value of An Annuity01234AAAAA（1+i）0A（1+i）1A（1+i）2A（1+i）3FCopyright 2001 Prentice-Hall,Inc.36 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
30、.Copyright 2001 Prentice-Hall,Inc.37Present Value of An Annuity 01234AAAAA/（1+i）1A/（1+i）2A/（1+i）3A/（1+i）4PCopyright 2001 Prentice-Hall,Inc.38Present Value of An Annuity P=A(1+i)-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)
31、.Refer to Table IV.Copyright 2001 Prentice-Hall,Inc.39Example-AnnuityYou are purchasing 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
32、 2001 Prentice-Hall,Inc.40Hint 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.Copyright 2001 Prentice-H
33、all,Inc.41PerpetuitieslThe 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+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.42 Julie Miller will receive the set o
34、f cash flows below.What is the Present Value at a discount rate of 10%?0 1 2 3 4$600$600$400$400$100PV0Mixed Flows ExampleCopyright 2001 Prentice-Hall,Inc.431.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 annuity stre
35、ams and any single cash flow group.Then discount each group back to t=0.How to Solve?Copyright 2001 Prentice-Hall,Inc.44 0 1 2 3 4 5$600$600$400$400$10010%$545.45$495.87$300.53$273.21$62.09$1677.15=PV0“Piece-At-A-Time”Copyright 2001 Prentice-Hall,Inc.45 0 1 2 3 4 5$600$600$400$400$10010%$1,041.60$57
36、3.57$62.10$1,677.27=PV0“Group-At-A-Time”(#1)$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)=$62.10Copyright 2001 Prentice-Hall,Inc.46 0 1 2 3 4$400$400$400$400 0 1 2 3 4 5$100 0 1 2$200$200$1,268.00$347.20$62.10PV0 equals$1677.3
37、0.PlusPlus“Group-At-A-Time”(#2)Copyright 2001 Prentice-Hall,Inc.471.Read problem thoroughly2.Determine if it is a PV or FV problem3.Create a time line4.Put cash flows and arrows on time line5.Determine if solution involves a single CF,annuity stream(s),or mixed flow6.Solve the problem7.Check with fi
38、nancial calculator(optional)Step To Solve Time Value Of Money ProblemsCopyright 2001 Prentice-Hall,Inc.48l General Formula:FVn=(1+i/m)mnn:Number of Yearsm:Compounding Periods per Yeari:Annual Interest RateFVn,m:FV at the end of Year nPV0:PV of the Cash Flow todayFrequency of CompoundingCopyright 200
39、1 Prentice-Hall,Inc.49Find“m”Annualized quotation basisInterest/periodLength of a periodAnnually compoundedr1 yearSemi-annually compoundedr/26 monthsQuarterly compoundedr/43 monthsMonthly compoundedr/121 monthWeekly compoundedr/521 week Daily compoundedr/3651 dayContinuously compounded Copyright 200
40、1 Prentice-Hall,Inc.50 example Julie Miller has$1,000 to invest for 2 years at an annual interest rate of 12%.?Annual FV2=1,000(1+.12/1)(1)(2)=1,254.40 Semi FV2 =1,000(1+.12/2)(2)(2)=1,262.48Impact of FrequencyCopyright 2001 Prentice-Hall,Inc.51Quarterly FV2=1,000(1+.12/4)(4)(2)=1,266.77Monthly FV2=
41、1,000(1+.12/12)(12)(2)=1,269.73Daily FV2=1,000(1+.12/365)(365)(2)=1,271.20Impact of FrequencyCopyright 2001 Prentice-Hall,Inc.52Calculation of EAR1 -mNIR 1 EARmlEffective annual interest rate（实际利率）:interest rate that is annualized using compound interest.lnominal interest rate（名义利率）:interest rate th
42、at is annualized using simple interest.lWhen interest is compounded more frequently than annually,the EAR will be greater than the NIR.Copyright 2001 Prentice-Hall,Inc.53Effective Interest Rates exampleGiven a monthly rate of 1%,what is the Effective Annual Rate(EAR)?What is the Annual Percentage Ra
43、te(APR)?EAR=(1+.01)12 1=.1268 or 12.68%APR=.01x12=.12 or 12%Copyright 2001 Prentice-Hall,Inc.54Comparing EARSlConsider the following interest rates quoted by three banks:Bank A:15%,compounded dailyBank B:15.5%,compounded quarterlyBank C:16%,compounded annuallyCopyright 2001 Prentice-Hall,Inc.55Compa
44、ring EARS16%1 -10.16 1 EAR16.42%1 -40.155 1 EAR16.18%1 -3650.15 1 EAR1CBank 4Bank B365ABank Copyright 2001 Prentice-Hall,Inc.56Comparing EARSlWhich is the best bank?For a saver,Bank B offers the best(highest)interest rate.For a borrower,Bank C offers the best(lowest)interest rate.lThe highest NIR is
45、 not necessarily the best.lCompounding during the year can lead to a significant difference between the NIR and the EAR.Copyright 2001 Prentice-Hall,Inc.57Types of LoanslA pure discount loan is a loan where the borrower receives money today and repays a single lump sum in the future.lAn interest onl
46、y loan requires the borrower to pay interest each period and to repay the entire principal本金 at some point in the future.lAn amortized loan requires the borrower to repay both the principal and interest over time.Copyright 2001 Prentice-Hall,Inc.581.Calculate the payment per period.2.Determine the i
47、nterest in Period t.(Loan balance借款余额 at t-1)x(i%/m)3.Compute principal payment债务总额in Period t.(Payment-interest from Step 2)4.Determine ending balance期末余额 in Period t.(Balance-principal payment from Step 3)5.Start again at Step 2 and repeat.Steps to Amortizing a LoanCopyright 2001 Prentice-Hall,Inc
48、.59 Julie Miller is borrowing$10,000 at a compound annual interest rate of 12%.Amortize the loan if annual payments are made for 5 years.Step 1:Payment PV0=R(PVIFA i%,n)$10,000=R(PVIFA 12%,5)$10,000=R(3.605)R=$10,000/3.605=$2,774Amortizing a Loan ExampleCopyright 2001 Prentice-Hall,Inc.60End of Year
49、 Payment Interest Principal Ending Balance 0-$10,000 1$2,774$1,200$1,574 8,426 2 2,774 1,011 1,763 6,663 3 2,774 800 1,974 4,689 4 2,774 563 2,211 2,478 5 2,775 297 2,478 0$13,871$3,871$10,000 Last Payment Slightly Higher Due to RoundingAmortizing a Loan Example 结束语当你尽了自己的最大努力时，失败也是伟大的，所以不要放弃，坚持就是正确的。When You Do Your Best,Failure Is Great,So DonT Give Up,Stick To The End谢谢大家荣幸这一路，与你同行ItS An Honor To Walk With You All The Way演讲人：XXXXXX 时 间：XX年XX月XX日
