Chapter 02 How to Calculate Present Values Lecture

Topics Covered 2-2 • Future Values and Present Values • Looking for Shortcuts—Perpetuities and Annuities • More Shortcuts—Growing Perpetuities and Annuities • How Interest Is Paid and Quoted Present Value and Future Value 2-3 Present Value Value today of a future cash flow. Future Value Amount to which an investment will grow after earning interest Future Values 2-4 Future Value of $100 = FV FV = $ 100 × (1 + r ) t Future Values 2-5 FV = $100 × (1 + r) t Example - FV What is the future value of $100 if interest is compounded annually at a rate of 7% for two years? FV = $100 × (1.07) × (1.07) = 114 .49 2 FV = $100 × (1 + .07) = $ 114 .49 Future Values with Compounding 2-6 1800 1600 1400 FV of $100 1200 0% 5% 10% 15% Interest Rates 1000 800 600 400 200 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Years Present Value 2-7 Present value = PV PV = discount factor × C1 Present Value 2-8 Discount factor = DF = PV of $1 DF = 1 (1 + r ) t Discount factors can be used to compute the present value of any cash flow Present Value 2-9 • The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. Also, you can reverse the prior example. PV = DF2 × C2 PV = (1+.107 ) 2 ×114 .49 = 100 Present Values with Compounding 2-10 120 100 Interest Rates PV of $100 80 0% 5% 10% 15% 60 40 20 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of Years Valuing an Office Building 2-11 Step 1: Forecast cash flows Cost of building = C0 = 700,000 Sale price in Year 1 = C1 = 800,000 Step 2: Estimate opportunity cost of capital If equally risky investments in the capital market offer a return of 7%, then Cost of capital = r = 7% Valuing an Office Building 2-12 Step 3: Discount future cash flows PV = C1 (1+ r ) = 800 , 000 (1+ .07 ) = 747 ,664 Step 4: Go ahead if PV of payoff exceeds investment NPV = 747 ,664 − 700 ,000 = 47 ,664 Net Present Value 2-13 NPV = PV - required investment NPV = C0 + C1 1+ r Risk and Present Value 2-14 • Higher risk projects require a higher rate of return • Higher required rates of return cause lower PVs PV of C 1 = $800,000 at 7% PV = 800,000 = 747 ,664 + 1 .07 Risk and Present Value 2-15 PV of C1 = $800,000 at 12% PV = 800,000 = 714 , 286 1 + .12 PV of C1 = $800,000 at 7% 800,000 = 747 ,664 PV = 1 + .07 Risk and Net Present Value 2-16 NPV = PV - required investment NPV = 714,286 - 700,000 = $14,286 Net Present Value Rule 2-17 • Accept investments that have positive net present value Example Use the original example. Should we accept the project given a 10% expected return? NPV = -700,000 + 800,000 = $27,273 1.10 Rate of Return Rule 2-18 • Accept investments that offer rates of return in excess of their opportunity cost of capital Example In the project listed below, the foregone investment opportunity is 12%. Should we do the project? Return = profit 800,000 − 700,000 = = .143 or 14.3% investment 700,000 Multiple Cash Flows 2-19 For multiple periods we have the discounted cash flow (DCF) formula PV0 = C1 (1+r )1 + C2 (1+ r ) 2 + ....+ T NPV0 = C 0 + ∑ (1+ rt ) t t =1 C Ct (1+ r ) t Net Present Values 2-20 $30,000 $ 870,000 Present value Year 0 0 -$700,000 30,000/1.12 = $26,786 870,000/1.122 = $693,559 Total = $20,344 1 2 Year Shortcuts 2-21 • Sometimes there are shortcuts that make it very easy to calculate the present value of an asset that pays off in different periods. These tools allow us to cut through the calculations quickly. Shortcuts 2-22 Perpetuity - Financial concept in which a cash flow is theoretically received forever. cash flow present value C r= PV Return = Shortcuts 2-23 Perpetuity - Financial concept in which a cash flow is theoretically received forever. cash flow discount rate C PV0 = 1 r PV of cash flow = Present Values 2-24 Example What is the present value of $1 billion every year, for all eternity, if you estimate the perpetual discount rate to be 10%? bil = $10 billion PV = $10.10 Present Values 2-25 Example - continued What if the investment does not start making money for 3 years? PV = $1 bil 0 . 10 × ( ) = $ 7 .51 billion 1 1.10 3 How to Value Annuities 2-26  1 1 − t r r (1 + r)    PV of annuity = C ×  Annuity - An asset that pays a fixed sum each year for a specified number of years Perpetuities & Annuities 2-27 PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years PVAF = 1 r − 1 r (1+ r ) t Short Cuts 2-28 Annuity - An asset that pays a fixed sum each year for a specified number of years. Asset Year of Payment 1 Perpetuity (first payment in year 1) 2…..t t+ 1 Present Value C r Perpetuity (first payment in year t + 1) C  1   r (1 + r )t Annuity from year 1 to year t C  C  1     −   t  r  r (1 + r )  Costing an Installment Plan 2-29 Example Tiburon Autos offers you “easy payments” of $5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car? 5,000 Present Value at year 0 5,000 5,000 5,000 5,000 Year 0 5,000 / 1.07 = 4,673 2 5,000 /(1.07) = 4,367 3 5,000 / (1.07) = 4,081 4 5,000 /(1.07) = 3,814 5 5,000 / (1.07) = 3,565 Total NPV = 20,501 1 2 3 4 5 Winning Big at the Lottery 2-30 Value = $357.5 million  1 1 − .036 .036(1 + .036) 30    Lottery value =19.683 ×  Example The state lottery advertises a jackpot prize of $590.5 million, paid in 30 installments over 30 years of $19.683 million per year, at the end of each year. If interest rates are 3.6% what is the true value of the lottery prize? Annuity Due 2-31 Annuity due - Level stream of cash flows starting immediately How does it differ from an ordinary annuity? PVAnnuity due = PVAnnuity × (1 + r ) How does the future value differ from an ordinary annuity? FVAnnuity due = FVAnnuity × (1 + r ) Annuities Due: Example 2-32 FVAD = FVAnnuity × (1 + r ) Example: Suppose you invest $429.59 annually at the beginning of each year at 10% interest. After 50 years, how much would your investment be worth?   550,003.81 50 Paying Off a Bank Loan 2-33 Example - Annuity You are purchasing a TV for $1,000. You are scheduled to make 4 annual installments. Given a rate of interest of 10%, what is the annual payment? $1,000 = PMT × PMT = $ 315 .47 [ 1 .10 ] − .10 (+1 1.10 )4 FV Annuity Short Cut 2-34 −1  (1 +r )t r    FV of annuity = C ×  Future Value of an Annuity – The future value of an asset that pays a fixed sum each year for a specified number of years. FV Annuity Short Cut 2-35 Example What is the future value of $20,000 paid at the end of each of the following 5 years, assuming your investment returns 8% per year? (1 +.08)5 −1 FV = 20,000 ×   .08  = $ 117 ,332 Constant Growth Perpetuity 2-36 PV0 = C1 r −g g = the annual growth rate of the cash flow Constant Growth Perpetuity 2-37 NOTE: This formula can be used to value a perpetuity at any point in time. PV0 = C1 r −g C t +1 PVt = r −g Constant Growth Perpetuity 2-38 Example What is the present value of $1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and a constant growth rate of 4%? 1 .10 − .04 = $16.667 billion PV0 = Effective Interest Rates 2-39 Effective Annual Interest Rate - Interest rate that is annualized using compound interest Annual Percentage Rate - Interest rate that is annualized using simple interest EAR & APR Calculations 2-40 Annual Percentage Rate (APR): APR = MR ×12 Effective Annual Interest Rate (EAR): EAR = (1 + MR )12 − 1 *where MR = monthly interest rate Effective Interest Rates 2-41 Example: Given a monthly rate of 1%, what is the effective annual rate (EAR)? What is the annual percentage rate (APR)? Effective Interest Rates 2-42 Example: Given a monthly rate of 1%, what is the effective annual rate (EAR)? What is the annual percentage rate (APR)? 12 EAR = (1 + .01) -1 = r EAR = (1 + .01) 12 -1 = .1268 or 12.68% APR = .01 × 12 = .12 or 12.00%