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1、ChapterMcGraw-Hill/IrwinCopyright 2006 by The McGraw-Hill Companies,Inc.All rights reserved.6 Discounted Cash Flow Discounted Cash Flow ValuationValuationKey Concepts and SkillsBe able to compute the future value of multiple cash flowsBe able to compute the present value of multiple cash flowsBe abl

2、e to compute loan paymentsBe able to find the interest rate on a loanUnderstand how interest rates are quotedUnderstand how loans are amortized or paid off1Chapter OutlineFuture and Present Values of Multiple Cash FlowsValuing Level Cash Flows:Annuities and PerpetuitiesComparing Rates:The Effect of

3、CompoundingLoan Types and Loan Amortization2Multiple Cash Flows Future Value Example 6.1Find the value at year 3 of each cash flow and add them together.Today(year 0 CF):3 N;8 I/Y;-7000 PV;CPT FV=8817.98Year 1 CF:2 N;8 I/Y;-4000 PV;CPT FV=4665.60Year 2 CF:1 N;8 I/Y;-4000 PV;CPT FV=4320Year 3 CF:valu

4、e=4,000Total value in 3 years=8817.98+4665.60+4320+4000=21,803.58Value at year 4:1 N;8 I/Y;-21803.58 PV;CPT FV=23,547.873Multiple Cash Flows FV Example 2Suppose you invest$500 in a mutual fund today and$600 in one year.If the fund pays 9%annually,how much will you have in two years?Year 0 CF:2 N;-50

5、0 PV;9 I/Y;CPT FV=594.05Year 1 CF:1 N;-600 PV;9 I/Y;CPT FV=654.00Total FV=594.05+654.00=1248.054Multiple Cash Flows Example 2 ContinuedHow much will you have in 5 years if you make no further deposits?First way:Year 0 CF:5 N;-500 PV;9 I/Y;CPT FV=769.31Year 1 CF:4 N;-600 PV;9 I/Y;CPT FV=846.95Total F

6、V=769.31+846.95=1616.26Second way use value at year 2:3 N;-1248.05 PV;9 I/Y;CPT FV=1616.265Multiple Cash Flows FV Example 3Suppose you plan to deposit$100 into an account in one year and$300 into the account in three years.How much will be in the account in five years if the interest rate is 8%?Year

7、 1 CF:4 N;-100 PV;8 I/Y;CPT FV=136.05Year 3 CF:2 N;-300 PV;8 I/Y;CPT FV=349.92Total FV=136.05+349.92=485.976Multiple Cash Flows Present Value Example 6.3Find the PV of each cash flows and add themYear 1 CF:N=1;I/Y=12;FV=200;CPT PV=-178.57Year 2 CF:N=2;I/Y=12;FV=400;CPT PV=-318.88Year 3 CF:N=3;I/Y=12

8、;FV=600;CPT PV=-427.07Year 4 CF:N=4;I/Y=12;FV=800;CPT PV=-508.41Total PV=178.57+318.88+427.07+508.41=1432.937Example 6.3 Timeline01234200400600800178.57318.88427.07508.411432.938Multiple Cash Flows Using a SpreadsheetYou can use the PV or FV functions in Excel to find the present value or future val

9、ue of a set of cash flowsSetting the data up is half the battle if it is set up properly,then you can just copy the formulasClick on the Excel icon for an example9Multiple Cash Flows PV Another ExampleYou are considering an investment that will pay you$1000 in one year,$2000 in two years and$3000 in

10、 three years.If you want to earn 10%on your money,how much would you be willing to pay?N=1;I/Y=10;FV=1000;CPT PV=-909.09N=2;I/Y=10;FV=2000;CPT PV=-1652.89N=3;I/Y=10;FV=3000;CPT PV=-2253.94PV=909.09+1652.89+2253.94=4815.9210非等額現金之計算非等額現金之計算 1112Multiple Uneven Cash Flows Using the CalculatorAnother w

11、ay to use the financial calculator for uneven cash flows is to use the cash flow keysPress CF and enter the cash flows beginning with year 0.You have to press the“Enter”key for each cash flowUse the down arrow key to move to the next cash flowThe“F”is the number of times a given cash flow occurs in

12、consecutive periodsUse the NPV key to compute the present value by entering the interest rate for I,pressing the down arrow and then computeClear the cash flow keys by pressing CF and then CLR Work13Decisions,DecisionsYour broker calls you and tells you that he has this great investment opportunity.

13、If you invest$100 today,you will receive$40 in one year and$75 in two years.If you require a 15%return on investments of this risk,should you take the investment?Use the CF keys to compute the value of the investmentCF;CF0=0;C01=40;F01=1;C02=75;F02=1NPV;I=15;CPT NPV=91.49No the broker is charging mo

14、re than you would be willing to pay.14Saving For RetirementYou are offered the opportunity to put some money away for retirement.You will receive five annual payments of$25,000 each beginning in 40 years.How much would you be willing to invest today if you desire an interest rate of 12%?Use cash flo

15、w keys:CF;CF0=0;C01=0;F01=39;C02=25000;F02=5;NPV;I=12;CPT NPV=1084.7115Saving For Retirement Timeline0 1 2 39 40 41 42 43 44 0 0 0 0 25K 25K 25K 25K 25KNotice that the year 0 cash flow=0(CF0=0)The cash flows years 1 39 are 0(C01=0;F01=39)The cash flows years 40 44 are 25,000(C02=25,000;F02=5)16Quick

16、 Quiz Part ISuppose you are looking at the following possible cash flows:Year 1 CF=$100;Years 2 and 3 CFs=$200;Years 4 and 5 CFs=$300.The required discount rate is 7%What is the value of the cash flows at year 5?What is the value of the cash flows today?What is the value of the cash flows at year 3?

17、17Annuities and Perpetuities DefinedAnnuity finite series of equal payments that occur at regular intervalsIf the first payment occurs at the end of the period,it is called an ordinary annuityIf the first payment occurs at the beginning of the period,it is called an annuity duePerpetuity infinite se

18、ries of equal payments18年 金年金為一特定期間內，定期支付的等額現金流量。年金的開始支付時點在第1期期末者稱為普通年金，在第1期期初者稱為期初年金。19期初年金普通年金1,000 1,0001,0001,0001,0001,0001,0001,0001,0001,000普通年金與期初年金20年金之未來值與現值年金之未來值與現值 一般年金示意圖一般年金示意圖 21一般金年未來值一般金年未來值2223一般金年現值一般金年現值24252627Annuities and Perpetuities Basic FormulasPerpetuity:PV=C/rAnnuities:

19、28Annuities and the CalculatorYou can use the PMT key on the calculator for the equal paymentThe sign convention still holdsOrdinary annuity versus annuity dueYou can switch your calculator between the two types by using the 2nd BGN 2nd Set on the TI BA-II Plus If you see“BGN”or“Begin”in the display

20、 of your calculator,you have it set for an annuity dueMost problems are ordinary annuities29Annuity Example 6.5You borrow money TODAY so you need to compute the present value.48 N;1 I/Y;-632 PMT;CPT PV=23,999.54($24,000)Formula:30Annuity Sweepstakes ExampleSuppose you win the Publishers Clearinghous

21、e$10 million sweepstakes.The money is paid in equal annual installments of$333,333.33 over 30 years.If the appropriate discount rate is 5%,how much is the sweepstakes actually worth today?30 N;5 I/Y;333,333.33 PMT;CPT PV=5,124,150.2931Buying a HouseYou are ready to buy a house and you have$20,000 fo

22、r a down payment and closing costs.Closing costs are estimated to be 4%of the loan value.You have an annual salary of$36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28%of your monthly income.The interest rate on the loan is 6%per year with monthly compounding(.5

23、%per month)for a 30-year fixed rate loan.How much money will the bank loan you?How much can you offer for the house?32Buying a House-ContinuedBank loanMonthly income=36,000/12=3,000Maximum payment=.28(3,000)=84030*12=360 N.5 I/Y840 PMTCPT PV=140,105Total PriceClosing costs=.04(140,105)=5,604Down pay

24、ment=20,000 5604=14,396Total Price=140,105+14,396=154,50133Annuities on the Spreadsheet-ExampleThe present value and future value formulas in a spreadsheet include a place for annuity paymentsClick on the Excel icon to see an example34Quick Quiz Part IIYou know the payment amount for a loan and you

25、want to know how much was borrowed.Do you compute a present value or a future value?You want to receive 5000 per month in retirement.If you can earn.75%per month and you expect to need the income for 25 years,how much do you need to have in your account at retirement?35Finding the PaymentSuppose you

26、 want to borrow$20,000 for a new car.You can borrow at 8%per year,compounded monthly(8/12=.66667%per month).If you take a 4 year loan,what is your monthly payment?4(12)=48 N;20,000 PV;.66667 I/Y;CPT PMT=488.2636Finding the Payment on a SpreadsheetAnother TVM formula that can be found in a spreadshee

27、t is the payment formulaPMT(rate,nper,pv,fv)The same sign convention holds as for the PV and FV formulasClick on the Excel icon for an example37Finding the Number of Payments Example 6.6The sign convention matters!1.5 I/Y1000 PV-20 PMTCPT N=93.111 MONTHS=7.75 yearsAnd this is only if you dont charge

28、 anything more on the card!38Finding the Number of Payments Another ExampleSuppose you borrow$2000 at 5%and you are going to make annual payments of$734.42.How long before you pay off the loan?Sign convention matters!5 I/Y2000 PV-734.42 PMT CPT N=3 years39Finding the RateSuppose you borrow$10,000 fr

29、om your parents to buy a car.You agree to pay$207.58 per month for 60 months.What is the monthly interest rate?Sign convention matters!60 N10,000 PV-207.58 PMTCPT I/Y=.75%40Annuity Finding the Rate Without a Financial CalculatorTrial and Error ProcessChoose an interest rate and compute the PV of the

30、 payments based on this rateCompare the computed PV with the actual loan amountIf the computed PV loan amount,then the interest rate is too lowIf the computed PV loan amount,then the interest rate is too highAdjust the rate and repeat the process until the computed PV and the loan amount are equal41

31、Quick Quiz Part IIIYou want to receive$5000 per month for the next 5 years.How much would you need to deposit today if you can earn.75%per month?What monthly rate would you need to earn if you only have$200,000 to deposit?Suppose you have$200,000 to deposit and can earn.75%per month.How many months

32、could you receive the$5000 payment?How much could you receive every month for 5 years?42Future Values for AnnuitiesSuppose you begin saving for your retirement by depositing$2000 per year in an IRA.If the interest rate is 7.5%,how much will you have in 40 years?Remember the sign convention!40 N7.5 I

33、/Y-2000 PMTCPT FV=454,513.0443Annuity DueYou are saving for a new house and you put$10,000 per year in an account paying 8%.The first payment is made today.How much will you have at the end of 3 years?2nd BGN 2nd Set(you should see BGN in the display)3 N-10,000 PMT8 I/YCPT FV=35,061.122nd BGN 2nd Se

34、t(be sure to change it back to an ordinary annuity)44Annuity Due Timeline 0 1 2 310000 10000 1000032,46435,016.1245Perpetuity Example 6.7Perpetuity formula:PV=C/rCurrent required return:40=1/rr=.025 or 2.5%per quarterDividend for new preferred:100=C/.025C=2.50 per quarter46Quick Quiz Part IVYou want

35、 to have$1 million to use for retirement in 35 years.If you can earn 1%per month,how much do you need to deposit on a monthly basis if the first payment is made in one month?What if the first payment is made today?You are considering preferred stock that pays a quarterly dividend of$1.50.If your des

36、ired return is 3%per quarter,how much would you be willing to pay?47Work the Web ExampleAnother online financial calculator can be found at MoneyChimpClick on the web surfer and work the following exampleChoose calculator and then annuity You just inherited$5 million.If you can earn 6%on your money,

37、how much can you withdraw each year for the next 40 years?Money chimp assumes annuity due!Payment=$313,497.8148Table 6.249Effective Annual Rate(EAR)This is the actual rate paid(or received)after accounting for compounding that occurs during the yearIf you want to compare two alternative investments

38、with different compounding periods you need to compute the EAR and use that for comparison.50Annual Percentage RateThis is the annual rate that is quoted by lawBy definition APR=period rate times the number of periods per yearConsequently,to get the period rate we rearrange the APR equation:Period r

39、ate=APR/number of periods per yearYou should NEVER divide the effective rate by the number of periods per year it will NOT give you the period rate51Computing APRsWhat is the APR if the monthly rate is.5%?.5(12)=6%What is the APR if the semiannual rate is.5%?.5(2)=1%What is the monthly rate if the A

40、PR is 12%with monthly compounding?12/12=1%Can you divide the above APR by 2 to get the semiannual rate?NO!You need an APR based on semiannual compounding to find the semiannual rate.52Things to RememberYou ALWAYS need to make sure that the interest rate and the time period match.If you are looking a

41、t annual periods,you need an annual rate.If you are looking at monthly periods,you need a monthly rate.If you have an APR based on monthly compounding,you have to use monthly periods for lump sums,or adjust the interest rate appropriately if you have payments other than monthly53Computing EARs-Examp

42、leSuppose you can earn 1%per month on$1 invested today.What is the APR?1(12)=12%How much are you effectively earning?FV=1(1.01)12=1.1268Rate=(1.1268 1)/1=.1268=12.68%Suppose if you put it in another account,you earn 3%per quarter.What is the APR?3(4)=12%How much are you effectively earning?FV=1(1.03

43、)4 =1.1255Rate=(1.1255 1)/1=.1255=12.55%54EAR-FormulaRemember that the APR is the quoted ratem is the number of compounding periods per year55Decisions,Decisions IIYou are looking at two savings accounts.One pays 5.25%,with daily compounding.The other pays 5.3%with semiannual compounding.Which accou

44、nt should you use?First account:EAR=(1+.0525/365)365 1=5.39%Second account:EAR=(1+.053/2)2 1=5.37%Which account should you choose and why?56Decisions,Decisions II ContinuedLets verify the choice.Suppose you invest$100 in each account.How much will you have in each account in one year?First Account:3

45、65 N;5.25/365=.014383562 I/Y;100 PV;CPT FV=105.39Second Account:2 N;5.3/2=2.65 I/Y;100 PV;CPT FV=105.37You have more money in the first account.57Computing APRs from EARs If you have an effective rate,how can you compute the APR?Rearrange the EAR equation and you get:58APR-ExampleSuppose you want to

46、 earn an effective rate of 12%and you are looking at an account that compounds on a monthly basis.What APR must they pay?59Computing Payments with APRsSuppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments.The entire computer system co

47、sts$3500.The loan period is for 2 years and the interest rate is 16.9%with monthly compounding.What is your monthly payment?2(12)=24 N;16.9/12=1.408333333 I/Y;3500 PV;CPT PMT=-172.8860Future Values with Monthly CompoundingSuppose you deposit$50 a month into an account that has an APR of 9%,based on

48、monthly compounding.How much will you have in the account in 35 years?35(12)=420 N9/12=.75 I/Y50 PMTCPT FV=147,089.2261Present Value with Daily CompoundingYou need$15,000 in 3 years for a new car.If you can deposit money into an account that pays an APR of 5.5%based on daily compounding,how much wou

49、ld you need to deposit?3(365)=1095 N5.5/365=.015068493 I/Y15,000 FVCPT PV=-12,718.5662Continuous CompoundingSometimes investments or loans are figured based on continuous compoundingEAR=eq 1The e is a special function on the calculator normally denoted by exExample:What is the effective annual rate

50、of 7%compounded continuously?EAR=e.07 1=.0725 or 7.25%63Quick Quiz Part VWhat is the definition of an APR?What is the effective annual rate?Which rate should you use to compare alternative investments or loans?Which rate do you need to use in the time value of money calculations?64Pure Discount Loan