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1、Chapter 3Time Value of MoneyTime Value of Money The Interest RateThe Interest Rate Simple InterestSimple Interest Compound InterestCompound Interest Compounding More Than Once Compounding More Than Once per Yearper Year Amortizing a LoanAmortizing a LoanChapter OutlineChapter OutlineBasic Definition
2、Present ValueValue today of a future cash flow.Future ValueAmount to which an investment will grow after earning interestBasic DefinitionDiscount RateInterest rate used to compute present values of future cash flows.Discount FactorPresent value of a$1 future payment.Obviously,$10,000 today.The reaso
3、n is that there is TIME VALUE OF MONEY!Which would you prefer-$10,000 today or$10,000 in 5 years?The Interest RateThe Interest RateTIME allows you the opportunity to postpone consumption and earn INTEREST.Why is TIME such an important element in your decision?Why TIME?Why TIME?uuCompound InterestInt
4、erest paid(earned)on any previous interest earned,as well as on the principal borrowed(lent).Interest on InterestTypes of InterestTypes of InterestuuSimple InterestInterest paid(earned)on only the original amount,or principal,borrowed(lent).FormulaFormulaSI=P0(i)(n)SI:Simple InterestP0:Deposit today
5、(t=0)i:Interest Rate per Periodn:Number of Time PeriodsSimple Interest FormulaSimple Interest FormulaSI=P0(i)(n)=$1,000(.07)(2)=$140Assume 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?Simple Interest ExampleSim
6、ple Interest ExampleFV=P0+SI=$1,000+$140 =$1,140 Future Value is the value at some future time of a present amount of money,or a series of payments,evaluated at a given interest rate.What is the Future Value(FV)of the deposit?Simple Interest(FV)Simple Interest(FV)The Present Value is simply the$1,00
7、0 you originally deposited.That is the value today!P0 =FV-SI Present Value is the current value of a future amount of money,or a series of payments,evaluated at a given interest rate.What is the Present Value(PV)of the previous problem?Simple Interest(PV)Simple Interest(PV)Why Compound Interest?Why
8、Compound Interest?Future Value(U.S.Dollars)Assume that you deposit$1,000 at a compound interest rate of 7%for 2 years.Future ValueFuture ValueSingle Deposit(Graphic)Single Deposit(Graphic)0 1 2$1,000FV27%FV1=P0(1+i)1 =$1,000(1.07)=$1,070Compound InterestYou earned$70 interest on your$1,000 deposit o
9、ver the first year.This is the same amount of interest you would earn under simple interest.Future ValueFuture ValueSingle Deposit(Formula)Single Deposit(Formula)FV1=P0(1+i)1FV2=P0(1+i)2General Future Value Formula:FVn=P0(1+i)n or FVn=P0(FVIFi,n)See Table IGeneral Future Value General Future Value F
10、ormulaFormulaFVIFi,n is on Table I at the end of the bookValuation Using Table IValuation Using Table IJulie Miller wants to know how large her deposit of$10,000$10,000 today will become at a compound annual interest rate of 10%for 5 years5 years.ExampleExample 0 1 2 3 4 5$10,000FV510%Calculation ba
11、sed on Table I:FV5=$10,000(FVIF10%,5)=$10,000(1.611)=$16,110 Due to RoundingStory Problem SolutionStory Problem SolutionuCalculation based on general formula:FVn=P0(1+i)n FV5=$10,000(1+0.10)5=$16,105.10We will use the“Rule-of-72”Quick!How long does it take to double$5,000 at a compound rate of 12%pe
12、r year(approx.)?Double Your Money!Double Your Money!a)72/12%=6 Years or b)72/6 years=12%Assume that you need$1,000 in 2 years.Lets examine the process to determine how much you need to deposit today at a discount rate of 7%compounded annually.Present ValuePresent Value Single Deposit(Graphic)Single
13、Deposit(Graphic)0 1 2$1,0007%PV1PV0General Future Value Formula:FVn=PV0(1+i)n FVn=PV0(FVIFi,n)General Present Value Formula:PV0=FVn/(1+i)n or PV0=FVn(PVIFi,n)-See Table 2General Present Value General Present Value FormulaFormulaPVIFi,n is on Table II at the end of the bookValuation Using Table IIVal
14、uation Using Table IIJulie Miller wants to know how large of a deposit to make so that the money will grow to$10,000 in 5 years at a discount rate of 10%.Story Problem ExampleStory Problem Example 0 1 2 3 4 5$10,00010%Calculation based on general formula:PV0=FVn/(1+i)n PV0=$10,000/(1+0.10)5=$6,209.2
15、1Calculation based on Table 2:PV0=$10,000(PVIF10%,5)=$10,000(.621)=$6,210.00 Due to RoundingStory Problem SolutionStory Problem SolutionIf you invest$1,000 today,you will receive$3,000 in exactly 8 years.What is the compound interest rate implicit in this situation?FVn=PV0(FVIFi,n)(FVIFi,8)=FV8/PV0
16、=3,000/1,000 =3 i=14.68%Unknown Interest RateHow long would it take for an investment of$1,000 to grow to$1,900 if we invested it at a compound annual interest rate of 10 percent?FVn=PV0(FVIFi,n)(FVIF10%,n)=FVn/PV0 =1,900/1,000 =1.9 n=6.72 yearsUnknown Number of Compounding PeriodsWhat is the future
17、 value of$1million invested at 10 percent for 25 years?FVn=PV0(FVIFi,n)FV25=PV0(FVIF10%,25)=$1,000,000*10.835 =$10,835,000Quick Quiz You need$30,000 in cash to buy a house 4 years from today.You expect to earn 5 percent on your savings.How much do you need to deposit today if this is the only money
18、you save for this purpose?PV0=FVn(PVIFi,n)PV0=FV4(PVIF5%,4)=$30,000*0.823 =$246,900Your firm has been told that it needs$74,300 today to fund a$120,000 expense 6 years from now.What rate of interest was used in the computation?FVn=PV0(FVIFi,n)(FVIFi,6)=FV6/PV0 =120,000/74,300 =1.615i=8.3%Ordinary An
19、nuity:Payments or receipts occur at the end of each period.Annuity Due:Payments or receipts occur at the beginning of each period.Types of AnnuitiesTypes of AnnuitiesuuAn Annuity represents a series of equal payments(or receipts)occurring over a specified number of equal distant periods.Student Loan
20、 Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement SavingsExamples of AnnuitiesOrdinary Annuity Ordinary Annuity 0 1 2 3$100$100$100EndEnd ofPeriod 1EndEnd ofPeriod 2TodayEqual Cash Flows Each 1 Period ApartEndEnd ofPeriod 3Annuity DueAnnuity Due0 1 2 3$100$100$100BeginningB
21、eginning ofPeriod 1BeginningBeginning ofPeriod 2TodayEqual Cash Flows Each 1 Period ApartBeginningBeginning ofPeriod 3It is an ordinary annuity whose payments or receipts continue forever.PVA=R/IThe ABS Co.wants to offer preferred stock for sale at a price of$60 a share.If the company wants their in
22、vestors to earn at least a 7.5 percent rate of return,what is the minimum annual dividend they will need to pay per share?R=PVA *i=60*7.5%=$4.5Perpetuity FVAn=R(FVIFAi,n)PVAn=R(PVIFAi,n)FVADn=R(FVIFAi,n)(1+i)PVADn=(1+i)(R)(PVIFAi,n)R is the period receiptFormula about Annuities What is the future va
23、lue of annual payments of$6,500 for eight years at 12 percent?FVAn=R(FVIFAi,n)FVA8=$6,500*(FVIFA12%,8)=$6,500*12.30 =$79,950Example of Annuit1.Read problem thoroughly2.Create a time line3.Put cash flows and arrows on time line4.Determine if it is a PV or FV problem5.Determine if solution involves a
24、single CF,annuity stream(s),or mixed flow6.Solve the problem7.Check with financial calculator(optional)Steps to Solve Time Value Steps to Solve Time Value of Money Problemsof Money ProblemsJulie Miller will receive the set of cash flows below.What is the Present Value at a discount rate of 10%.Mixed
25、 Flows ExampleMixed Flows Example 0 1 2 3 4 5$600$600$400$400$100PV010%1.Solve a“piece-at-a-time”by discounting each piece back to t=0.2.Solve a“group-at-a-time”by firstbreaking problem into groups of annuity streams and any single cash flow groups.Then discount each group back to t=0.How to Solve?H
26、ow to Solve?“Piece-At-A-Time”Piece-At-A-Time”0 1 2 3 4 5$600$600$400$400$10010%$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$62.09$62.09$1677.15$1677.15=PVPV0 0 of the Mixed Flowof the Mixed Flow“Group-At-A-Time”(#1)Group-At-A-Time”(#1)0 1 2 3 4 5$600$600$400$400$100$600$600$400$400$10010
27、%$1,041.60$1,041.60$573.57$573.57$62.10$62.10$1,677.27$1,677.27 =PVPV0 0 of Mixed Flow of Mixed Flow Using TablesUsing Tables$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)=$62.10“Group-At-A-Time”(#2)Group-At-A-Time”(#2)0 1 2 3
28、4$400$400$400$400$400$400$400$400PV0 equals$1677.30.0 1 2$200$200$200$200 0 1 2 3 4 5$100$100$1,268.00$1,268.00$347.20$347.20$62.10$62.10PlusPlusJulie Miller will receive the set of cash flows below.What is the Present Value at a discount rate of 7%.(PIECE AT A TIME AND GROUP AT A TIME)SOLVE PROBLEM
29、SOLVE PROBLEM 0 1 2 3 4 5$100$500$500$500$5007%44Nominal Interest RateThis is the annual rate that is quoted by law that has not been adjusted for frequency of compounding.If interest is compounded more that once a year,the effective interest rate will be higher that the nominal rate.45Effective Ann
30、ual Rate(EAR)This is the actual rate paid(or received)after adjusting for compounding that occurs during the yearIf you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.46Compounding More than Once a YearFuture Val
31、ues with Monthly CompoundingSuppose you deposit$50 a month into an account that has an interest rate of 12%,based on monthly compounding.How much will you have in the account in 1 years?FVFVn n=P0(1+i/m)mn PV PV0 0=FVFVn n/(1+i/m)mn FVFV1 1=$50(1+12%/12)12*1=$50*1.127=$56.3547Continuous CompoundingS
32、ometimes investments or loans are figured based on continuous compoundingFVn=PV0(e)in 48Things to RememberYou ALWAYS need to make sure that the interest rate and the time period match.If you are looking at annual periods,you need an annual rate.If you are looking at monthly periods,you need a monthl
33、y rate.49EAR-FormulaRemember that the i is the quoted ratem is the number of compounding periods per yearIf a savings plan offered a nominal interest rate of 8%compounded quarterly on a one-year investment,what s the effective annual interest rate?EAR=1+(8%/4)4-1=8.243%Example Loan Types and Loan Am
34、ortizationPure discount loansA borrower receives money today and repays a single lump sum at some time in the future.Interest-only loansA borrower pays interest each period and repays the entire principle at some point in the future.Amortized loansA lender requires the borrower to repay parts of the
35、 loan amount over time.2023/5/9Essentials of Corporate Finance51The process of paying off a loan by making regular principle reductions is called amortizing the loan.Installment-type LoanIt is repaid in equal periodic payments that include both interest and principle.These payments can be made month
36、ly,quarterly,semiannually,or annually.Common examples:mortgage loans,auto loans,consumer loans,and certain business loans.Amortizing a Loan1.Calculate the payment per period.2.Determine the interest in Period t.(Loan Balance at t-1)x(i%/m)3.Compute principal payment in Period t.(Payment-Interest fro
37、m Step 2)4.Determine ending balance in Period t.(Balance-principal payment from Step 3)5.Start again at Step 2 and repeat.Steps to Amortizing a LoanSteps to Amortizing a LoanJulie Miller is borrowing$10,000 at a compound annual interest rate of 12%.Amortize the loan if annual payments are made for 5
38、 years.Step 1:Payment on Table IV 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 ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleAmortizing a Loan ExampleLast Payment Slightly Higher Due to Rounding1.Understand what is meant by the time valu
39、e of money.2.Understand the relationship between present and future value.3.Describe how the interest rate can be used to adjust the value of cash flows both forward and backward to a single point in time.4.Calculate both the future and present value of:(a)an amount invested today;(b)a stream of equ
40、al cash flows(an annuity);and(c)a stream of mixed cash flows.5.Distinguish between an“ordinary annuity”and an“annuity due.”6.Use interest factor tables and understand how they provide a shortcut to calculating present and future values.7.Use interest factor tables to find an unknown interest rate or
41、 growth rate when the number of time periods and future and present values are known.8.Build an“amortization schedule”for an installment-style loan.After studying Chapter 3,you should be able to:57Comprehensive ProblemYou have$10,000 to invest for five years.1.How much is the interest on interest wh
42、en the investment provides 10%annual return?2.How much additional interest will you earn if the investment provides a 5%annual return,when compared to a 4%annual return?3.How long will it take your$10,000 to double in value if it earns 5%annually?4.What annual rate has been earned if$1,000 grows into$4,000 in 20 years?Page 67-Q12