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1、Discounted Cash Flow ValuationChapter 4Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinAppreciate the significance of compound vs. simple interestDescribe and compute the future value and/or present value of a single cash flow or series of cash flowsDefine and
2、calculate the return on an investmentRecognize and compute the impact of compounding periods on the true return of stated interest ratesDevelop facility with a financial calculator and/or spreadsheet to solve time value problemsComprehend and calculate time value metrics for perpetuities and annuiti
3、esFamiliarization with loan types and amortizationKey Concepts and SkillsKey Concepts and Skills4-14.1 Valuation: The One-Period Case4.2 The Multiperiod Case4.3 Compounding Periods4.4 Simplifications4.5 Loan Types and Loan Amortization4.6 What Is a Firm Worth?Chapter OutlineChapter Outline4-2A dolla
4、r today is more valuable than a dollar to be received in the futureWhy?A dollar today is more valuable because:It can be invested to make more dollarsIt can be immediately consumedThere is no doubt about its receiptThe Essential PremiseThe Essential Premise4-3If you know your required rate of return
5、 and the length of time before cash is harvested, you can calculate some critical metrics:The value today of a payment to be received in the futureThis measure is called a “Present Value”The value in the future of a sum invested todayThis measure is called a “Future Value”Present and Future Values c
6、an be calculated over single and multiple periodsAn Important CorollaryAn Important Corollary4-4If you were to invest $10,000 at 5-percent interest for one year, 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 to
7、tal 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). 4.1 The One-Period Case4.1 The One-Period Case4-5In the one-period case, the formula for FV can be written as:FV = C0(1 + r)Where C0 is cash flow today (time zero
8、), and r is the appropriate interest rate.Future ValueFuture Value4-6Present Value is todays value of a sum to be received in the future given a specific rate of interest and time horizon.Suppose you were promised $10,000 due in one year when interest rates are 5-percent. Your investment be worth $9
9、,523.81 in todays dollars. Present ValuePresent ValueThe 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 the Present Value (PV).Note that $10,000 = $9,523.81(1.05).4-7In the one-period case, the formula for PV can be written as:P
10、resent ValuePresent ValueWhere C1 is cash flow at date 1, and r is the appropriate interest rate.4-8The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment.Suppose an investment that promises to pay $10,000 in one year is offered
11、for sale for $9,500. Your interest rate is 5%. Should you buy?Net Present ValueNet Present Value4-9Net Present ValueNet Present ValueThe present value of the cash inflow is greaterthan the cost. In other words, the Net PresentValue is positive, so the investment should be purchased.4-10In the one-pe
12、riod case, the formula for NPV can be written as:NPV = Cost + PVNet Present ValueNet Present ValueIf we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5 percent, our FV would be less than the $10,000 the investment promised, and
13、 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.Future Value and CompoundingFuture Value and Compounding4-17Future Value and CompoundingFuture Value and Compounding0123454-18Multiperiod Present ValueMultiperiod Present ValueoThe general f
14、ormula for the present value of an investment over many periods can be written as:PV = CT / (1 + r)TWhere CT is cash flow at date T,r is the appropriate interest rate, andT is the number of periods over which the cash is invested.4-19How much would an investor have to set aside today in order to hav
15、e $20,000 five years from now if the current rate is 15%?Example: Multiperiod Present Value Example: Multiperiod Present Value and Discountingand Discounting012345$20,000PV4-20Examples thus far have offered the time and interest rate and solved for PV or FVKeep in mind that there are four variables:
16、PVFVTRIf you have any three you can solve for the fourthThe math can become cumbersomeFinancial Calculators and Spreadsheets are very helpfulSolve for Any VariableSolve for Any Variable4-21If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?Example: Solving
17、for TimeExample: Solving for Time4-22Assume the total cost of a college education will be $50,000 when your child enters college in 12 years. You have $5,000 to invest today. What rate of interest must you earn on your investment to cover the cost of your childs education? Example: Solving for Requi
18、red Example: Solving for Required ReturnReturnAbout 21.15%.4-23Texas Instruments BA-II PlusFV = future valuePV = present valueI/Y = periodic interest rateP/Y must equal 1 for the I/Y to be the periodic rateInterest is entered as a percent, not a decimalN = number of periodsRemember to clear the regi
19、sters (CLR TVM) after each problemOther calculators are similar in formatCalculator Orientation: KeysCalculator Orientation: Keys4-24Consider 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 present v
20、alue of this stream of cash flows?If the issuer offers this investment for $1,500, should you purchase it?Multiple Cash FlowsMultiple Cash Flows4-25Multiple Cash FlowsMultiple Cash Flows01234200400600800178.57318.88427.07508.411,432.93Present Value Cost Do Not Purchase4-26First, set your calculator
21、to 1 payment per year.Then, use the cash flow menu:Valuing Uneven Cash FlowsValuing Uneven Cash FlowsCF2CF1F2F1CF0120011,432.930400INPV12CF4CF3F4F3160018004-27All examples thus far have assumed annual compoundingInstances of other compounding schedules abound:Banks compound interest quarterly, month
22、ly or dailyMortgage companies compound interest monthlyYet, almost all interest rates are expressed annuallyIf a rate is expressed annually, but compounded more frequently, then the effective rate is higher than the stated rateThis concept is called the Effective Annual Rate or EAR4.3 Compounding Pe
23、riods4.3 Compounding Periods4-28Compounding an investment m times a year for T years provides for the future value of wealth:Computing FV with Multiple Compounding Computing FV with Multiple Compounding Computing FV with Multiple Compounding Computing FV with Multiple Compounding PeriodsPeriodsPerio
24、dsPeriods4-29Compounding PeriodsCompounding PeriodsqFor example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to4-30A reasonable question to ask in the above example is “what is the effective annual rate of interest on that investment?”Effective Annual Rat
25、es of Effective Annual Rates of InterestInterestThe Effective Annual Rate (EAR) of interest is the annual rate that would give us the same end-of-investment wealth after 3 years:4-31So, investing at 12.36% compounded annually is the same as investing at 12% compounded semi-annually.Effective Annual
26、Rates of Effective Annual Rates of InterestInterest4-32Find the Effective Annual Rate (EAR) of an 18% APR loan that is compounded monthly.What we have is a loan with a monthly interest rate rate of 1%.This is equivalent to a loan with an annual interest rate of 19.56%.Effective Effective Annual Annu
27、al Rates Rates of of InterestInterest4-33EAR on a Financial CalculatorEAR 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 year4-34The general formula for the fu
28、ture 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 constant that is approximately equal to 2.718. ex is a key on your calculator.Continuous Compoun
29、dingContinuous Compounding4-35PerpetuityA constant stream of cash flows that lasts foreverGrowing perpetuityA stream of cash flows that grows at a constant rate foreverAnnuityA stream of constant cash flows that lasts for a fixed number of periodsGrowing annuityA stream of cash flows that grows at a
30、 constant rate for a fixed number of periods4.4 Simplifications4.4 Simplifications4-36A constant stream of cash flows that lasts foreverPerpetuityPerpetuity01C2C3C4-37What is the value of a British consol that promises to pay 15 every year for ever? The interest rate is 10%.Perpetuity: ExamplePerpet
31、uity: Example01152153154-38A growing stream of cash flows that lasts foreverGrowing PerpetuityGrowing Perpetuity01C2C(1+g)3C (1+g)24-39The expected dividend next 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 s
32、tream?Growing Perpetuity: ExampleGrowing Perpetuity: Example01$1.302$1.30(1.05)3$1.30 (1.05)24-40A constant stream of cash flows with a fixed maturityAnnuityAnnuity01C2C3CTC4-41If you can afford a $400 monthly car payment, how much car can you afford if interest rates are 7% on 36-month loans?Annuit
33、y: ExampleAnnuity: Example01$4002$4003$40036$4004-4243What 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%?0 1 2 3 4 5$100 $100 $100 $100$323.97$297.22A growing stream of cash flows with a fixed maturityGrowing
34、 AnnuityGrowing Annuity01C2C(1+g)3C (1+g)2T C(1+g)T-14-44A 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 value at retirement if the discount rate is 10 percent?Growing Annuity: ExampleGrowin
35、g Annuity: Example01$20,0002$20,000(1.03)40 $20,000(1.03)394-45Growing Annuity: ExampleGrowing Annuity: ExampleYou are evaluating an income generating property. Net rent is received 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.
36、 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 5$34,706.264-46Pure Discount Loans are the simplest form of loan. The borrower receives money today and repays a single lump sum (principal and interest) at a future time.Interest-O
37、nly Loans require an interest payment each period, with full principal due at maturity.Amortized Loans require repayment of principal over time, in addition to required interest.4.5 Loan Types and Loan 4.5 Loan Types and Loan AmortizationAmortization4-47Treasury bills are excellent examples of pure
38、discount loans. The principal amount is repaid at some future date, without any periodic interest payments.If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?PV = 10,000 / 1.07 = 9,345.79Pure Discount LoansPure
39、Discount Loans4-48Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.What would the stream of cash flows be?Years 1 4: Interest payments of .07(10,000) = 700Year 5: Interest + principal = 10,700This cash flow stream is similar to
40、 the cash flows on corporate bonds, and we will talk about them in greater detail later.Interest-Only LoanInterest-Only Loan4-49Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.Click on the Excel ic
41、on to see the amortization tableAmortized Loan with Fixed Amortized Loan with Fixed Principal PaymentPrincipal Payment4-50Each payment covers the interest expense plus reduces principalConsider a 4 year loan with annual payments. The interest rate is 8% ,and the principal amount is $5,000.What is th
42、e annual payment?4 N8 I/Y5,000 PVCPT PMT = -1,509.60Click on the Excel icon to see the amortization tableAmortized Loan with Fixed Amortized Loan with Fixed PaymentPayment4-51Conceptually, a firm should be worth the present value of the firms cash flows.The tricky part is determining the size, timin
43、g and risk of those cash flows.4.6 What Is a Firm Worth?4.6 What Is a Firm Worth?4-52You can solve time value problems in any of four ways:Math (Formulae given above)Tables (See Appendix A)Financial CalculatorSpreadsheet SoftwareFinancial calculators and spreadsheet software are the most common meth
44、ods now.A Note About TechniqueA Note About Technique4-53How is the future value of a single cash flow computed?How is the present value of a series of cash flows computed.What is the Net Present Value of an investment?What is an EAR, and how is it computed?What is a perpetuity? An annuity?Contrast interest-only loans to amortized loans.Quick QuizQuick Quiz4-54
