7-1 CHAPTER 7 Time Value of Money Future value Present value Rates of return Amortization Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-2 Time lines show timing of cash flows. 0 1 2 3 CF1 CF2 CF3 i% CF0 Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-3 Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Year i% 100 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-4 Time line for an ordinary annuity of $100 for 3 years. 0 1 2 3 100 100 100 i% Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-5 Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3. 0 1 2 3 100 75 50 i% -50 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-6 What’s the FV of an initial $100 after 3 years if i = 10%? 0 1 2 3 10% 100 FV = ? Finding FVs is compounding. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-7 After 1 year: FV1 = PV + INT1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00. After 2 years: FV2 = PV(1 + i)2 = $100(1.10)2 = $121.00. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-8 After 3 years: FV3 = PV(1 + i)3 = $100(1.10)3 = $133.10. In general, FVn = PV(1 + i)n. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7-9 Four Ways to Find FVs Solve the equation with a regular calculator. Use tables. Use a financial calculator. Use a spreadsheet. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 10 Algebraic Solution FVn = PV(1 + i)n. FV3 = 100(1 + .10) 3 FV3 = 100(1.331) = 133.10 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 11 Solution Using Tables FVn = PV(FVIF i,n). FV3 = 100(FVIF 10%, 3) Use FVIF table from pages A-6 & 7, Table A3 FV3 = 100(1.331) = 133.10 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 12 Financial Calculator Solution Financial calculators solve this equation: FVn = PV(1 + i)n. There are 4 variables. If 3 are known, the calculator will solve for the 4th. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 13 Here’s the setup to find FV: INPUTS 3 N 10 I/YR -100 PV 0 PMT OUTPUT FV 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1, END Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 14 What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding. 0 1 2 3 10% PV = ? Copyright © 2002 by Harcourt, Inc. 100 All rights reserved. 7 - 15 Solve FVn = PV(1 + i )n for PV: FVn PV = (1 + i)n = FVn ( 1 PV = $100 1.10 ) 3 ( ) 1 n . 1+i = $100(PVIFi,n) Table A1 = $100(0.7513) = $75.13. So the (PVIF 10%,3) = .7513 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 16 Financial Calculator Solution INPUTS 3 N OUTPUT 10 I/YR PV -75.13 0 PMT 100 FV Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 17 If sales grow at 20% per year, how long before sales double? Solve for n: FVn = $1(1 + i)n; n $2 = $1(1.20) ln 2 = ln 1.2n .693147=.18232n 3.801 = n Use calculator to solve, see next slide. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 18 If sales grow at 20% per year, how long before sales double? Solve for n: FVn = PV(FVIF i,n); $2 = $1(FVIF 20,n) 2.00 = (FVIF 20,n) n between 3 and 4 years 1.728 and 2.0736 Use calculator to solve, see next slide. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 19 INPUTS OUTPUT N 3.8 20 I/YR -1 PV 0 PMT 2 FV Graphical Illustration: FV 2 3.8 1 Year 0 1 Copyright © 2002 by Harcourt, Inc. 2 3 4 All rights reserved. 7 - 20 Compound Growth How do you find the compound growth rate for your company to analyze sales growth ? Can use either PV or FV formula, use FV 1062021 (1 + i )9 = 5284371 (1 + i )9 = 5284371/1062021 (1 + i)9 = 4.976 (1 + i) = 4.976 .111 (1 + i) = 1.195 i = .195 or 19.5% Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 21 Tabular Solution Use PV formula and table A-1 5284371 (PVIF i,9) = 1062021 PVIF i,9 = 1062021/5284371 PVIF i,9 = .20097 Use table A-3, for 9 Periods, find .20097 i is between 18% and 20% Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 22 Calculator Solution, Compound Growth INPUTS 9 -1062021 0 N OUTPUT I/YR PV PMT 5284371 FV 19.51 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 23 What’s the difference between an ordinary annuity and an annuity due? Ordinary Annuity 0 i% 1 2 3 PMT PMT PMT 1 2 3 PMT PMT Annuity Due 0 i% PMT Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 24 What’s the FV of a 3-year ordinary annuity of $100 at 10%? 0 1 2 100 100 3 10% Copyright © 2002 by Harcourt, Inc. 100 110 121 FV = 331 All rights reserved. 7 - 25 Algebraic Solution FVA =( PMT)* ( 1 +i)n – 1 I FVA =( 100)* ( 1 + .1)3 – 1 .1 FVA =( 100)* 1.331 – 1 = 100 * 3.31 = .1 FVA = 331.00 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 26 Tabular Solution FVA i,n =( PMT) * (FVIFA i,n ) Use Table A-4 on pages A-8 & 9 FVA 10%,3 =( 100) * (FVIFA 10%,3) FVA 10%,3 =( 100)* 3.31 = FVA 10%,3 = 331.00 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 27 Financial Calculator Solution INPUTS 3 10 0 -100 N I/YR PV PMT OUTPUT FV 331.00 Have payments but no lump sum PV, so enter 0 for present value. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 28 What’s the PV of this ordinary annuity? 0 1 2 3 100 100 100 10% 90.91 82.64 75.13 248.68 = PV Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 29 Algebraic Solution 1 . PVA =( PMT)* 1 – (1 + i)n i 1 . PVA =( 100) * 1 – (1 + .1)3 .1 PVA =( 100)* 1 - .7513 = 100 * 2.48685 = .1 PVA = 248.69 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 30 Tabular Solution PVA i,n =( PMT) * (PVIFA i,n ) Use Table A-2 on pages A-4 & 5 PVA 10%,3 =( 100) * (PVIFA 10%,3) PVA 10%,3 =( 100)* 2.4869 = PVA 10%,3 = 248.69 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 31 INPUTS 3 10 N I/YR OUTPUT PV 100 0 PMT FV -248.69 Have payments but no lump sum FV, so enter 0 for future value. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 32 Find the FV and PV if the annuity were an annuity due. 0 1 2 100 100 3 10% 100 Easiest way, multiply results by (1 + i). Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 33 Algebraic Solution 1 . PVAD i,n =( PMT)* 1 – (1 + i)n * (1 + i) i 1 . PVAD 10%,3 =( 100) * 1 – (1 + .1)3 * (1 + .1) .1 PVAD 10%,3 =( 100)* 1 - .7513 * (1.1) = .1 100 * 2.48685 * (1.1)= PVAD 10%,3 = 273.55 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 34 Tabular Solution PVAD i,n =( PMT) * (PVIFA i,n )* (1 + i) Use Table A-2 on pages A-4 & 5 PVAD10%,3 =( 100) * (PVIFA 10%,3) * (1 + i) PVAD 10%,3 =( 100)* 2.4869 * 1.1 = PVAD 10%,3 = 273.55 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 35 Switch from “End” to “Begin.” Then enter variables to find PVA3 = $273.55. INPUTS 3 10 N I/YR OUTPUT PV 100 0 PMT FV -273.55 Then enter PV = 0 and press FV to find FV = $364.10. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 36 What is the PV of this uneven cash flow stream? 0 1 2 3 4 100 300 300 -50 10% 90.91 247.93 225.39 -34.15 530.08 = PV Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 37 Input in “CFLO” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50 Enter I = 10, then press NPV button to get NPV = $530.09. (Here NPV = PV.) Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 38 The Power of Compound Interest A 20-year old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095) in an online stock account. The stock account has an expected annual return of 12%. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 39 How much money by the age of 65? INPUTS 45 12 0 -1095 N I/YR PV PMT OUTPUT FV 1,487,261.89 If she begins saving today, and sticks to her plan, she will have $1,487,261.89 by the age of 65. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 40 How much would a 40-year old investor accumulate by this method? INPUTS 25 12 0 -1095 N I/YR PV PMT OUTPUT FV 146,000.59 Waiting until 40, the investor will only have $146,000.59, which is over $1.3 million less than if saving began at 20. So it pays to get started early. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 41 How much would the 40-year old investor need to save to accumulate as much as the 20-year old? INPUTS 25 12 0 N I/YR PV OUTPUT 1487261.89 PMT FV -11,154.42 The 40-year old investor would have to save $11,154.42 every year, or $30.56 per day to have as much as the investor beginning at the age of 20. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 42 Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 43 Rules for Non-annual Compounding 95% of the time, the method for adjusting for non-annual compounding is: Divide i by m, m being the # of compounding periods in a year. Multiply n by m Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 44 FV of $100 after 3 years under 10% semiannual compounding? Quarterly? iNom FVn = PV 1 + m FV3S FV3Q m*n 0.10 = $100 1 + 2 . 2*3 = $100(1.05)6 = $134.01. = $100(1.025)12 = $134.49. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 45 0 1 2 3 10% 100 133.10 Annually: FV3 = $100(1.10)3 = $133.10. 0 0 1 1 2 3 2 4 5 3 6 5% 100 134.01 Semiannually: FV6 = $100(1.05)6 = $134.01. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 46 Exam Question (Example) Your uncle has given you a choice between receiving $20,000 today on your 18th birthday, or waiting until your 25th birthday and receiving $40,000. If you would invest in a junk bond fund if you took the $20,000, expecting to average 10% per year, compounded semiannually, which would you prefer? Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 47 Exam Question (Example) See board for timeline. Algebraic solution: FV = PV(1 + i/m)n*m FV = 20,000 (1 + .1/2)7 * 2 FV = 20,000 ( 1.9799) = 39,598.63 Prefer the $40,000 in 7 years. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 48 Exam Question (Example) Tabular solution: PV = FV(PVIF i/2,n*2) PV = 40,000 (PVIF 10/2,7*2) Table A-1 PV = 40,000 ( .505) = 20,202 Prefer the $40,000 in 7 years (same conclusion. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 49 Financial Calculator Solution INPUTS 14 10 ? N I/YR PV OUTPUT 0 PMT 40,000 FV -20,202.72 Could have solved for FV inputting PV calcuation: P/Y set to 2 Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 50 We will deal with 3 different rates: iNom = nominal, or stated, or quoted, rate per year. iPer = periodic rate. effective annual EAR = EFF% = . rate Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 51 iNom is stated in contracts. Periods per year (m) must also be given. Examples: 8%; Quarterly 8%, Daily interest (365 days) Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 52 Periodic rate = iPer = iNom/m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: iPer = 8%/4 = 2%. 8% daily (365): iPer = 8%/365 = 0.021918%. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 53 Effective Annual Rate (EAR = EFF%): The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + iNom/m)m = (1.05)2 = 1.1025. EFF% = 10.25% because (1.1025)1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 54 An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 55 How do we find EFF% for a nominal rate of 10%, compounded semiannually? ( ) =(1 + 0.10 ) – 1.0 2 iNom m EFF = 1 + –1 m 2 = (1.05)2 – 1.0 = 0.1025 = 10.25%. Or use a financial calculator. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 56 EAR = EFF% of 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%. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 57 Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 58 When is each rate used? iNom: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 59 iPer: Used in calculations, shown on time lines. If iNom has annual compounding, then iPer = iNom/1 = iNom. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 60 EAR = EFF%: Used to compare returns on investments with different payments per year. (Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 61 What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? 0 1 2 3 4 5% 5 6 6-mos. periods 100 Copyright © 2002 by Harcourt, Inc. 100 100 All rights reserved. 7 - 62 Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 63 1st Method: Compound Each CF 0 5% 1 2 100 3 4 100 5 6 100.00 110.25 121.55 331.80 FVA3 = $100(1.05)4 + $100(1.05)2 + $100 = $331.80. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 64 2nd Method: Treat as an Annuity Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: EAR = ( 0.10 1+ 2 Copyright © 2002 by Harcourt, Inc. 2 ) – 1 = 10.25%. All rights reserved. 7 - 65 Or, to find EAR with a calculator: NOM% = 10. P/YR = 2. EFF% = 10.25. Copyright © 2002 by Harcourt, Inc. All rights reserved. 7 - 66 b. The cash flow stream is an annual annuity. Find kNom (annual) whose EFF% = 10.25%. In calculator, EFF% = 10.25 P/YR = 1 NOM% = 10.25 c. INPUTS 3 10.25 0 -100 N I/YR PV PMT OUTPUT Copyright © 2002 by Harcourt, Inc. FV 331.80 All rights reserved. 7 - 67 What’s the PV of this stream? 0 1 2 3 100 100 100 5% 90.70 82.27 74.62 247.59 Copyright © 2002 by Harcourt, Inc. All rights reserved.