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Time Value of Money Future Value and Compounding Present Value and Discounting Annuity and Perpetuity Amortization Loans Different Compounding Periods 1 Principle of Time Value Time is money. A dollar today is more valuable than a dollar tomorrow Why? Money can earn a positive rate of return at no risk. 2 Time Line 0 1 2 3 i% CF0 CF1 CF2 CF3 Time lines show the timing of cash flows. Tick marks are at the end of periods. Time 0 is today; Time 1 is placed at the end of Period 1 or at the beginning of Period 2. 3 Time line for a $100 cash flow received at the end of Year 2 0 1 2 Year i% 100 4 Time line for equal payments of $100 for 3 years 0 1 2 3 i% -100 -100 -100 5 Time line for an uneven cash flow stream 0 1 2 3 i% -150 100 75 50 6 Time Value of Money One of the most important fundamental concepts in finance. A dollar in hand today is worth more than a dollar to be received in the future. FV = PV (1 + i )n 7 What’s the value of an initial $100 deposit after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? The process of finding FVs is called compounding. 8 After 1 year: FV1 = PV + INT1 = PV + PV i = PV (1 + i ) = $100 (1.10) = $110.00 9 After 2 years: FV2 = FV1 (1 + i ) = 110 (1.10) = $100 (1.10)2 = PV (1 + i )2 = $121.00 10 After 3 years: FV3 =121(1 +0.10 ) =100(1 +0.10 )2 (1+0.10) =100(1.10)3 =PV(1 + i )3 =$133.10 In general: FVn = PV (1 + i )n 11 Future value equation FVn = PV (1 + i )n There are 4 variables in the equation: FVn , PV, i and n. If any 3 variables are known, the calculator will solve for the 4th. 12 The setup to find FV: INPUTS 3 10 100 0 N I/YR PV PMT FV OUTPUT -133.10 13 How much should you save now to have $100 in 3 years if i = 10%? 0 1 2 3 10% PV = ? 100 Finding PVs is discounting, and it’s the reverse of compounding. 14 Present Value Equation PV = FVn (1 + i )n 3 1 PV = $100 1.10 = $100 0.7513 = $75.13 15 Financial Calculator Solution INPUTS 3 10 0 100 N I/YR PV PMT FV OUTPUT -75.13 16 What interest rate would cause $100 to grow to $125.97 in 3 years? Solve for i : 100 (1 + i )3 = $125.97 INPUTS 3 -100 0 125.97 N I/YR PV PMT FV OUTPUT 8.0 17 If sales grow at 20% per year, how long before sales double? Solve for n: FVn = PV (1 + i )n 2 = 1 (1.20)n INPUTS 20 -1 0 2 N I/YR PV PMT FV OUTPUT 3.8 18 Definition of Annuity Annuity: a series of cash flows of an equal amount at fixed intervals for a specified number of periods. Ordinary Annuity: an annuity whose payments occur at the end of each period. Annuity Due: an annuity whose payments occur at the beginning of each period. 19 What’s the difference between an ordinary annuity and an annuity due? Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT 20 What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 3 10% 100 100 100 110 121 FV = 331 21 Financial Calculator Solution INPUTS 3 10 0 -100 N I/YR PV PMT FV OUTPUT 331.00 22 What’s the PV of this ordinary annuity? 0 1 2 3 10% 100 100 100 90.91 82.64 75.13 248.68 = PV 23 Financial Calculator Solution INPUTS 3 10 100 0 N I/YR PV PMT FV OUTPUT -248.69 24 Loan Analysis Loan amount=PV of all payments discounted at the contractual interest rate 25 Amortization Loans Amortized Loan: a loan that is repaid in equal payments over its life. Each periodic payment includes not only interest but also a portion of principal. 26 If you borrow a 3-year, $1,000, 10% amortized loan, how much do you need to pay every year? 0 1 2 3 10% 1,000 PMT PMT PMT INPUTS 3 10 1000 0 N I/YR PV PMT FV OUTPUT -402.11 27 Loan= $1,000 PV of all payments discounted at 10%=Loan of $,1000 0 1 2 3 10% 1,000 PMT PMT PMT INPUTS 3 10 1000 0 N I/YR PV PMT FV OUTPUT -402.11 28 Amortization Schedule Amortization Schedule is a table that shows precisely how a loan will be repaid. Construct a amortization schedule for the 3-year, $1,000, 10% amortized loan. 29 Find interest charge for Year 1 Interest charge = Initial balance interest rate = $1,000 (0.10) = $100 30 Find principal repayment in Year 1 Principal repayment = Total payment – interest charge = $402.11 - $100 = $302.11 31 Find end balance for Year 1 End balance = Initial balance - principal repayment = $1,000 - $302.11 = $697.89 Repeat these steps for Years 2 and 3 to complete the amortization table. 32 Amortization Schedule BEG PRIN END YR BAL PMT INT PMT BAL 1 $1,000 $402 $100 $302 $698 2 698 402 70 332 366 3 366 402 36 366 0 1,206 206 1,000 33 $ 402.11 Interest charge 302.11 Principal repayments 0 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. 34 Different Compounding Periods Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated i% constant? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often. 35 Annual vs. Semiannual Compounding 0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10 0 1 2 3 Year 0 1 2 3 4 5 6 Period 5% 100 ? 36 Annual vs. Semiannual Compounding INPUTS 6 5 100 0 N I/YR PV PMT FV OUTPUT -134.01 37 Effective Annual Rate EAR is the interest rate which causes PV to grow to the same FV as under multi-period compounding. EAR is the actual rate of return investors earn, or the actual rate of interest borrowers pay. 38 Effective Annual Rate The return on an investment with monthly payments is different from one with the same nominal value but quarterly payments. We must convert both into EAR basis to compare the rates of return. 39 Calculating Effective Annual Rate m iNom EAR = 1 + -1 m 0 . 10 2 = 1 + - 1 .0 2 = ( . 05 )2 - 1 . 0 1 = 0 . 1025 = 10 . 25 % 40 Calculating Effective Annual Rate Nominal Rate = 10% EARAnnual = 10% EARQ = (1 + 0.10/4)4 - 1 = 10.38% EARM = (1 + 0.10/12)12 - 1 = 10.47% EARD(365) = (1 + 0.10/365)365 - 1 = 10.52% 41 Calculating annual nominal rate (inom) from effective compounding period rate (keff) inom=keffxm Also, keff=inom/m 42 Perpetuities A perpetuity is an annuity that continues forever. The present value of a perpetuity is: PMT PV = i 43 Growing Perpetuities The cash flows of a growing perpetuity grow at a constant rate forever. The present value of a growing perpetuity is: PMT1 PV = i-g 44

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posted: | 6/30/2011 |

language: | English |

pages: | 44 |

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