1、McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Discounted Cash Flow ValuationChapter 4McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Key Concepts and SkillsoBe able to compute the future value and/or present value of a singl
2、e cash flow or series of cash flowsoBe able to compute the return on an investmentoBe able to use a financial calculator and/or spreadsheet to solve time value problemsoUnderstand perpetuities and annuitiesMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Chapter O
3、utline4.1 Valuation:The One-Period Case4.2 The Multiperiod Case4.3 Compounding Periods4.4 Simplifications4.5 What Is a Firm Worth?McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.4.1 The One-Period CaseoIf you were to invest$10,000 at 5-percent interest for one ye
4、ar,your investment would grow to$10,500.$500 would be interest($10,000 .05)$10,000 is the principal repayment($10,000 1)$10,500 is the total due.It can be calculated as:$10,500=$10,000(1.05)qThe total amount due at the end of the investment is call the Future Value(FV).McGraw-Hill/IrwinCopyright 200
5、7 by The McGraw-Hill Companies,Inc.All rights reserved.Future ValueoIn the one-period case,the formula for FV can be written as:FV=C0(1+r)TWhere C0 is cash flow today(time zero),and r is the appropriate interest rate.McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserve
6、d.Present ValueoIf you were to be promised$10,000 due in one year when interest rates are 5-percent,your investment would be worth$9,523.81 in todays dollars.05.1000,10$81.523,9$The amount that a borrower would need to set aside today to be able to meet the promised payment of$10,000 in one year is
7、called the Present Value(PV).Note that$10,000=$9,523.81(1.05).McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Present ValueoIn the one-period case,the formula for PV can be written as:rCPV11Where C1 is cash flow at date 1,and r is the appropriate interest rate.Mc
8、Graw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Net Present ValueoThe Net Present Value(NPV)of an investment is the present value of the expected cash flows,less the cost of the investment.oSuppose an investment that promises to pay$10,000 in one year is offered fo
9、r sale for$9,500.Your interest rate is 5%.Should you buy?McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Net Present Value81.23$81.523,9$500,9$05.1000,10$500,9$NPVNPVNPVThe present value of the cash inflow is greaterthan the cost.In other words,the Net PresentVal
10、ue is positive,so the investment should be purchased.McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Net Present ValueIn the one-period case,the formula for NPV can be written as:NPV=Cost+PVIf we had not undertaken the positive NPV project considered on the last
11、slide,and instead invested our$9,500 elsewhere at 5 percent,our FV would be less than the$10,000 the investment promised,and we would be worse off in FV terms:$9,500(1.05)=$9,975$1.10+5$1.10.40=$3.30This is due to compounding.McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All right
12、s reserved.Future Value and Compounding01234510.1$3)40.1(10.1$02.3$)40.1(10.1$54.1$2)40.1(10.1$16.2$5)40.1(10.1$92.5$4)40.1(10.1$23.4$McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Present Value and DiscountingoHow much would an investor have to set aside today
13、in order to have$20,000 five years from now if the current rate is 15%?012345$20,000PV5)15.1(000,20$53.943,9$McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.How Long is the Wait?If we deposit$5,000 today in an account paying 10%,how long does it take to grow to$1
14、0,000?TrCFV)1(0T)10.1(000,5$000,10$2000,5$000,10$)10.1(T)2ln()10.1ln(Tyears 27.70953.06931.0)10.1ln()2ln(TMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Assume the total cost of a college education will be$50,000 when your child enters college in 12 years.You ha
15、ve$5,000 to invest today.What rate of interest must you earn on your investment to cover the cost of your childs education?What Rate Is Enough?TrCFV)1(012)1(000,5$000,50$r10000,5$000,50$)1(12r12110)1(r2115.12115.1110121rAbout 21.15%.McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.Al
16、l rights reserved.Calculator KeysoTexas Instruments BA-II PlusnFV=future valuenPV=present valuenI/Y=periodic interest rateoP/Y must equal 1 for the I/Y to be the periodic rateoInterest is entered as a percent,not a decimalnN=number of periodsnRemember to clear the registers(CLR TVM)after each proble
17、mnOther calculators are similar in formatMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Multiple Cash FlowsoConsider an investment that pays$200 one year from now,with cash flows increasing by$200 per year through year 4.If the interest rate is 12%,what is the p
18、resent value of this stream of cash flows?oIf the issuer offers this investment for$1,500,should you purchase it?McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Multiple Cash Flows01234200400600800178.57318.88427.07508.411,432.93Present Value Cost Do Not Purchase
19、McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Valuing“Lumpy”Cash FlowsFirst,set your calculator to 1 payment per year.Then,use the cash flow menu:CF2CF1F2F1CF0120011,432.930400INPV12CF4CF3F4F316001800McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,
20、Inc.All rights reserved.4.3 Compounding PeriodsCompounding an investment m times a year for T years provides for future value of wealth:TmmrCFV10McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Compounding Periodsq For example,if you invest$50 for 3 years at 12%co
21、mpounded semi-annually,your investment will grow to93.70$)06.1(50$212.150$632FVMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Effective Annual Rates of InterestA reasonable question to ask in the above example is“what is the effective annual rate of interest on
22、that investment?”The Effective Annual Rate(EAR)of interest is the annual rate that would give us the same end-of-investment wealth after 3 years:93.70$)06.1(50$)212.1(50$632FV93.70$)1(50$3EARMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Effective Annual Rates o
23、f InterestSo,investing at 12.36%compounded annually is the same as investing at 12%compounded semi-annually.93.70$)1(50$3EARFV50$93.70$)1(3 EAR1236.150$93.70$31EARMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Effective Annual Rates of InterestoFind the Effectiv
24、e Annual Rate(EAR)of an 18%APR loan that is compounded monthly.oWhat we have is a loan with a monthly interest rate rate of 1%.oThis is equivalent to a loan with an annual interest rate of 19.56%.1956.1)015.1(1218.111212mnmrMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights
25、reserved.EAR on a Financial Calculatorkeys:description:2nd ICONVOpens interest rate conversion menu EFF=CPT19.56Texas Instruments BAII Plus NOM=18 ENTERSets 18 APR.C/Y=12 ENTERSets 12 payments per yearMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Continuous Com
26、poundingoThe general formula for the future value of an investment compounded continuously over many periods can be written as:FV=C0erTWhere C0 is cash flow at date 0,r is the stated annual interest rate,T is the number of years,ande is a transcendental number approximately equal to 2.718.ex is a ke
27、y on your calculator.McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.4.4 SimplificationsoPerpetuitynA constant stream of cash flows that lasts foreveroGrowing perpetuitynA stream of cash flows that grows at a constant rate foreveroAnnuitynA stream of constant cas
28、h flows that lasts for a fixed number of periodsoGrowing annuitynA stream of cash flows that grows at a constant rate for a fixed number of periodsMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.PerpetuityA constant stream of cash flows that lasts forever01C2C3C3
29、2)1()1()1(rCrCrCPVrCPV McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Perpetuity:ExampleWhat is the value of a British consol that promises to pay 15 every year for ever?The interest rate is 10-percent.011521531515010.15PVMcGraw-Hill/IrwinCopyright 2007 by The M
30、cGraw-Hill Companies,Inc.All rights reserved.Growing PerpetuityA growing stream of cash flows that lasts forever01C2C(1+g)3C(1+g)2322)1()1()1()1()1(rgCrgCrCPVgrCPVMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Growing Perpetuity:ExampleThe expected dividend next
31、 year is$1.30,and dividends are expected to grow at 5%forever.If the discount rate is 10%,what is the value of this promised dividend stream?01$1.302$1.30(1.05)3$1.30(1.05)200.26$05.10.30.1$PVMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.AnnuityA constant strea
32、m of cash flows with a fixed maturity01C2C3CTrCrCrCrCPV)1()1()1()1(32TrrCPV)1(11TCMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Annuity:ExampleIf you can afford a$400 monthly car payment,how much car can you afford if interest rates are 7%on 36-month loans?01$4
33、002$4003$40059.954,12$)1207.1(1112/07.400$36PV36$400McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.What is the present value of a four-year annuity of$100 per year that makes its first payment two years from today if the discount rate is 9%?22.297$09.197.327$0PV
34、0 1 2 3 4 5$100$100$100$100$323.97$297.2297.323$)09.1(100$)09.1(100$)09.1(100$)09.1(100$)09.1(100$4321411ttPVMcGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Growing AnnuityA growing stream of cash flows with a fixed maturity01CTTrgCrgCrCPV)1()1()1()1()1(12TrggrCP
35、V)1(112C(1+g)3C(1+g)2T C(1+g)T-1McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Growing Annuity:ExampleA defined-benefit retirement plan offers to pay$20,000 per year for 40 years and increase the annual payment by three-percent each year.What is the present valu
36、e at retirement if the discount rate is 10 percent?01$20,00057.121,265$10.103.1103.10.000,20$40PV2$20,000(1.03)40$20,000(1.03)39McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Growing Annuity:ExampleYou are evaluating an income generating property.Net rent is rec
37、eived at the end of each year.The first years rent is expected to be$8,500,and rent is expected to increase 7%each year.What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%?0 1 2 3 4 5500,8$)07.1(500,8$2)07.1(500,8$095,9$65.731,9$3)07.1(500,8$87
38、.412,10$4)07.1(500,8$77.141,11$34,706.26McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.4.5 What Is a Firm Worth?oConceptually,a firm should be worth the present value of the firms cash flows.oThe tricky part is determining the size,timing and risk of those cash
39、flows.McGraw-Hill/IrwinCopyright 2007 by The McGraw-Hill Companies,Inc.All rights reserved.Quick QuizoHow is the future value of a single cash flow computed?oHow is the present value of a series of cash flows computed.oWhat is the Net Present Value of an investment?oWhat is an EAR,and how is it computed?oWhat is a perpetuity?An annuity?
