Slide 1Future ValueFinding Present ValueNet Present ValueNet Present Value Decision RuleNet Present Value ProblemNet Present Value ProblemNet Present Value ProblemCompoundingAnother View of CompoundingCompounding ExampleCompounding ExampleIs it true?DiscountingLottery Value TodayEffect of Change in Discount RateCash Flow ExamplePresent Value of Cash FlowsValuing a Stream of Cash Flows IIValuing a Stream of Cash Flows IIStream of Cash Flows -CalculatorNumber of PeriodsCompounding PeriodsGeneral Compounding FormulaCompoundingEffective Annual RateAPR to EARAPR to EAR with CalculatorCompounding for several yearsChase Example ContinuedChase Compounding AnswerAPR and EARFinding EAR to Solve ProblemContinuous Compounding?PerpetuityPerpetuity ExamplePerpetuity ExampleGrowing PerpetuityGrowing PerpetuityBuilding ValueVERY IMPORTANT Note about PerpetuitiesStock Value as a Growing PerpetuityGordon Growth Model ExampleAnnuitiesLottery WinnerFuture Value of AnnuitiesLottery WinnerRetirement - FV of An AnnuityRetirement - FV of An AnnuityTwo Step AnnuityTwo Step Annuity ExampleAnnuity TypesAnnuity Due ExampleADD FUTURE VALUE ANNUITY DUEGrowing AnnuitiesGrowing Annuity ExampleGrowing Annuity Example SolutionInfrequent AnnuityInfrequent AnnuityInfrequent AnnuityAmortizationAmortization TableAmortizationSlide 64Firm ValuationAnother MethodGraphical Firm ValuationAmortization PracticeAmortization PracticeAnnuity Due Extra Calculator Trick© J. David Miller 2017Discounted Cash Flow ValuationChapter 4Finance 357© J. David Miller 2017Future ValueMost projects in the real world involve decisions about the future.Project:Cost $100,000 todayBenefit $105,000 in one yearWould you suggest that your company accept the project?2© J. David Miller 2017Finding Present ValueProject:Cost $100,000 todayBenefit $105,000 in one yearSuppose the interest rate was 7%?We can move the benefit back to today by discounting it to find the present value (PV).105,000 / (1 + 7%) = $98,130.84The net benefit today would be-$100,000 + $98,130.84 = -$1,869.16We call the net benefit today the net present value (NPV).3© J. David Miller 2017Net Present ValueThe net present value or NPV is the present value of cash flows we receive minus the present value of cash we must pay out.The book uses the formula NPV = -Cost + PVUsing the previous example, we were told that the cost was $100,000 today and we found that the present value that we would receive was $98,130.84.NPV = -$100,000 + $98,130.84 = -$1,869.164© J. David Miller 2017Net Present Value Decision RuleThe NPV Decision Rule says that we accept projects if they have a positive NPV.If we must decide between multiple projects, we accept the project with the greatest NPV.***This is a fundamental rule of finance***5© J. David Miller 2017Net Present Value ProblemA year ago, you developed an app that you believe will revolutionize the dry cleaning industry. You are already so busy managing the business, that you barely have time for school. The app is currently being used by 350 dry cleaning businesses. A larger mobile app company has offered to buy the business for $1,000,000 once it is being used by 500 businesses. At the rate it is growing, that will be at the end of two years. However, if you hired a professional manager who would cost $100,000 in salary, you believe that it would reach that level at the end of one year. Should you continue to manage the business yourself or hire the manager?6© J. David Miller 2017Net Present Value ProblemA year ago, you developed an app that you believe will revolutionize the dry cleaning industry. You are already so busy managing the business, that you barely have time for school. The app is currently being used by 350 dry cleaning businesses. A larger mobile app company has offered to buy the business for $1,000,000 once it is being used by 500 businesses. At the rate it is growing, that will be in two years. However, if you hired a professional manager who would cost $100,000 in salary, you believe that it would reach that level in one year. Interest rates are 6%. Should you continue to manage the business yourself or hire the manager?Manage the business yourself Hire a manager***** You should always draw a chart of your cash flows ******Year Cash Flow1 02 $1,000,000Year Cash Flow1 -$100,000 + $1,000,000 7© J. David Miller 2017Net Present Value ProblemInterest rate is 6%Manage the business yourself Hire a managerManage yourself Hire a managerNPV = 1,000,000 / (1.06)2 NPV = (-100,000 + 1,000,000)/(1.06)1 NPV = 1,000,000 / 1.1236 NPV = $900,000 / 1.06 NPV = $889,966.40 NPV = $849,056.60Disregarding your grades, it is best to manage the business yourself.Would this change if interest rates were 15%?Year Cash Flow1 02 $1,000,000Year Cash Flow1 -$100,000 + $1,000,000 8© J. David Miller 2017CompoundingThe process of leaving money in an investment and allowing it to grow over multiple periods is called compounding. We invest $1000 in a bank account that has an interest rate of 5% per year. At the end of year 1, it is worth $1050. This is called simple interest.Leaving the money in the account for the second year allows it to grow by 5% again. However, this time we are starting year two with $1050. We are earning interest on our previously earned interest.9© J. David Miller 2017Another View of Compounding Another way to accomplish this is to combine the steps. By combining the process into one step, we are calculating compound interest.Which equals the formula for a cash flow that is compounded over multiple time periods. T = the number of time periods over which the money is invested. 10© J. David Miller 2017Compounding ExampleA stock you own currently pays a dividend of $1.50 per share. The dividend is expected to grow by 20% per year for the next 5 years. How much will the dividend be 5 years from now?11© J. David Miller 2017Compounding ExampleA stock you own currently pays a dividend of $1.50 per share. The dividend is expected to grow by 20% per year for the next 5 years. How much will the dividend be 5 years from now?Method 1 - FormulaFV = $1.5 * (1 + .2)5 = $3.73Method 2 – Financial CalculatorPV = -1.5 N = 5 I/Y = 20 FV = ?FV = $3.7312© J. David Miller 2017Is it true?"Compound interest is the greatest force in the universe" Attributed to Albert Einstein13© J. David Miller 2017DiscountingWith a little simple algebra, the compounding formula turns into