PowerPoint PresentationKey Concepts and Skills4.1 The One-Period CaseFuture ValuePresent ValueSlide 6Net Present ValueSlide 84.2 The Multiperiod CaseSlide 10Future Value and CompoundingPresent Value and DiscountingSlide 13How Long is the Wait?What Rate Is Enough?Multiple Cash FlowsMultiple Cash FlowsSlide 18Slide 194.3 Compounding PeriodsCompounding PeriodsSlide 22Effective Annual Rates of InterestSlide 24Some TerminologiesContinuous Compounding4.4 SimplificationsPerpetuitySlide 29Slide 30Perpetuity: ExampleGrowing PerpetuitySlide 33Growing Perpetuity: ExampleAnnuitySlide 36Slide 37Annuity: ExampleSlide 39Growing AnnuityGrowing Annuity: ExampleSlide 42Loan AmortizationPure Discount LoansInterest-Only LoanAmortized Loan with Fixed Principal PaymentWhat Is a Firm Worth?McGraw-Hill/IrwinCopyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved CHAPTER4Lecture Note 5:Discounted Cash Flow ValuationSlide 2Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinKey Concepts and Skills•compute the future value and/or present value of series of cash flows•compute the return on an investment•use a spreadsheet to solve time value problems•Understand perpetuities and annuitiesSlide 3Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/Irwin4.1 The One-Period CaseIf 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 total due. It can be calculated as:$10,500 = $10,000×(1.05)The total amount due at the end of the investment is call the Future Value (FV).Slide 4Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinFuture Value•In the one-period case, the formula for FV can be written as:FV = C0×(1 + r)Where C0 is cash flow today (time zero), and r is the appropriate interest rate.Slide 5Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinPresent Value•If 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 today’s 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 called the Present Value (PV).Note that $10,000 = $9,523.81×(1.05).Slide 6Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinPresent Value•In the one-period case, the formula for PV can be written as:rCPV11Where C1 is cash flow at date 1, and r is the appropriate interest rate.Slide 7Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinNet Present Value•The Net Present Value (NPV) of an investment is the present value of the expected cash flows, less the cost of the investment:NPV = –Cost + PV•Suppose an investment that promises to pay $10,000 in one year is offered for sale for $9,500. Your interest rate is 5%. Should you buy?05.1000,10$500,9$ NPVSlide 8Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinNet Present Value81.23$81.523,9$500,9$ NPVThe present value of the cash inflow is greaterthan the cost. In other words, the Net PresentValue is positive, so the investment should be purchased.If 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 we would be worse off in FV terms: $9,500×(1.05) = $9,975 < $10,000Slide 9Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/Irwin4.2 The Multiperiod Case•The general formula for the future value of an investment over many periods can be written as:FV = C0×(1 + r)TWhere C0 is cash flow at date 0,r is the appropriate interest rate, andT is the number of periods over which the cash is invested. I.e., C2 = C0×(1 + r)2where (1 + r)2 = 1 + 2r + r2Slide 10Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinFuture Value•Suppose a stock currently pays a dividend of $1.10, which is expected to grow at 40% per year for the next five years.•What will the dividend be in five years?FV = C0×(1 + r)T$5.92 = $1.10×(1.40)5Slide 11Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinFuture Value and Compounding•Notice that the dividend in year five, $5.92, is considerably higher than the sum of the original dividend plus five increases of 40-percent on the original $1.10 dividend:$5.92 > $1.10 + 5×[$1.10×.40] = $3.30This is due to compounding.Slide 12Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinPresent Value and Discounting•The PV of a cash payment in period T discounted at interest rate rTTrCPV)1( Slide 13Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinPresent Value and Discounting•How much would an investor have to set aside today in order to have $20,000 five years from now if the current rate is 15%?0 1 2 3 4 5$20,000PV5)15.1(000,20$53.943,9$ Slide 14Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinHow Long is the Wait?If we deposit $5,000 today in an account paying 10%, how long does it take to grow to $10,000?TrCFV )1(0T)10.1(000,5$000,10$ 2000,5$000,10$)10.1( T)2ln()10.1ln( Tyears 27.70953.06931.0)10.1ln()2ln(TSlide 15Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinAssume 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 child’s education? What Rate Is Enough?TrCFV )1(012)1(000,5$000,50$ r10000,5$000,50$)1(12 r12110)1(  r2115.12115.1110121rAbout 21.15%.Slide 16Copyright © 2008 by The McGraw-Hill Companies, Inc. All rights reserved McGraw-Hill/IrwinMultiple Cash Flows•Consider 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 value of this