CHAPTER 6 Time Value of Money
6-1 Chapter 6 The Time Value of Money Future Value Present Value Rates of Return Amortization Copyright © 2000 by Harcourt, Inc. All rights reserved. Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 328876777. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-2 Cash Flow Time Lines Graphical representations used to show the timing of cash flows. 0 1 2 CF1 CF2 3 i% CF0 CF3 Tick marks at ends of periods, so t=0 is today; t=1 is the end of Period 1; or the beginning of Period 2. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-3 Time line for a $100 lump sum due at the end of Year 2. 0 1 2 Year i% 100 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-4 Time line for an ordinary annuity of $100 for 3 years. 0 1 2 100 100 3 i% 100 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-5 Time line for uneven CFs -$50 at t=0 and $100, $75, and $50 at the end of Years 1 through 3. 0 1 2 100 75 3 i% -50 50 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-6 Future Value The amount to which a cash flow or series of cash flows will grow over a period of time when compounded at a given interest rate. How much would you have at the end of one year if you deposited $100 in a bank account that pays 5 percent interest each year? FVn = FV1 = PV + INT = PV + (PV x i) = PV (1 + i) = $100(1+0.05) = $100(1.05) = $105 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-7 What’s the FV of an initial $100 after 3 years if i = 10%? 0 1 2 3 i=10% 100 FV = ? Finding FVs is Compounding. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-8 Future Value After 1 year: FV1 = PV + i = 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. After 3 years: FV3 = PV(1 + i)3 = 100 (1.10)3 = $133.10. In general, FVn = PV (1 + i)n Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-9 Three ways to find Future Values: Solve the Equation with a Regular Calculator Use Tables Use a Financial Calculator Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-10 Financial Calculator Solution: Financial calculators solve this equation: FV n PV 1 i . n There are 4 variables. If 3 are known, the calculator will solve for the 4th. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-11 Financial Calculator Solution: Here’s the setup to find FV: INPUTS 3 N 10 -100 I/YR PV OUTPUT 0 PMT FV 133.10 Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set: P/YR = 1 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-12 Present Value Present value is the value today of a future cash flow or series of cash flows. Discounting is the process of finding the present value of a cash flow or series of cash flows, the reverse of compounding. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-13 What is the PV of $100 due in 3 years if i = 10%? 0 1 2 3 10% PV = ? 100 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-14 What is the PV of $100 due in 3 years if i = 10%? Solve FVn = PV (1 + i )n for PV: n FV 1 n PV = = FV n 1+ i 1 + in 3 1 = $100 PVIF PV = $100 i,n 1.10 = $100 0.7513 = $75.13. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-15 Financial Calculator Solution: INPUTS OUTPUT 3 N 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 (C) 2000 by Harcourt, Inc. All rights reserved. 6-16 If sales grow at 20% per year, how long before sales double? Solve for n: FVn = 1(1 + i)n; 2 = 1(1.20)n . Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-17 Financial Calculator Solution: INPUTS OUTPUT Graphical Illustration: 2 20 I/YR N 3.8 -1 PV 0 PMT 2 FV FV 3.8 1 Year 0 1 2 3 4 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-18 Future Value of an Annuity Annuity: A series of payments of an equal amount at fixed intervals for a specified number of periods. Ordinary (deferred) 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. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-19 Ordinary Annuity Versus Annuity Due 0 i% 0 1 2 PMT PMT 1 2 PMT PMT 3 PMT 3 i% PMT Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-20 What’s the FV of a 3-year Ordinary Annuity of $100 at 10%? 0 1 2 100 100 3 10% 100 110 121 FV = 331 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-21 Financial Calculator Solution: INPUTS OUTPUT 3 10 0 -100 N I/YR PV PMT FV 331.00 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-22 Present Value of an Annuity PVAn = the present value of an annuity with n payments Each payment is discounted, and the sum of the discounted payments is the present value of the annuity Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-23 What is the PV of this Ordinary Annuity? 0 1 2 100 100 3 10% 100 90.91 82.64 75.13 248.69 = PV Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-24 Financial Calculator Solution: INPUTS 3 N OUTPUT 10 I/YR 100 PV PMT 0 FV -248.69 Have payments but no lump sum FV, so enter 0 for future value. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-25 Find the FV and PV if the Annuity were an Annuity Due. 0 1 2 100 100 3 10% 100 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-26 Financial Calculator Solution: Switch from “End” to “Begin”. Then enter variables to find PVA3 = $273.55. INPUTS OUTPUT 3 10 N I/YR 100 PV PMT 0 FV -273.55 Then enter PV = 0 and press FV to find FV = $364.10. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-27 Solving for Interest Rates with Annuities You pay $864.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of the year. What interest rate will you earn on this investment? 0 1 2 3 4 250 250 250 i=? - 864.80 250 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-28 Financial Calculator Solution: PVAn = PMT(PVIFAi,n) $846.80 = $250(PVIFA i = ?,4) INPUTS OUTPUT 4 ? N I/YR - 846.80 250 PV PMT 0 FV =7.0 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-29 Uneven Cash Flow Streams A series of cash flows in which the amount varies from one period to the next. Payment (PMT) designates constant cash flows Cash flow (CF) designates cash flows in general, including uneven cash flows Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-30 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 (C) 2000 by Harcourt, Inc. All rights reserved. 6-31 Financial Calculator Solution: Input in “Cash Flow” 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 (C) 2000 by Harcourt, Inc. All rights reserved. 6-32 What interest rate would cause $100 to grow to $125.97 in 3 years? $100 (1 + i )3 = $125.97. INPUTS 3 N OUTPUT -100 I/YR PV 0 PMT 125.97 FV 8% Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-33 Semiannual and Other Compounding Periods Annual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year. Semiannual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added twice a year. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-34 Will the FV of a lump sum be larger or smaller if we compound more often, holding the state i% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semi-annually, quarterly, or daily-interest is earned on interest more often. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-35 0 1 2 3 10% 100 133.10 Annually: FV3 = 100(1.10)3 = 133.10. 0 1 0 1 2 2 3 4 3 5 6 5% 100 134.01 Semi-annually: FV6/2 = 100(1.05)6 = 134.01. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-36 Distinguishing Between Different Interest Rates iSIMPLE = Simple (Quoted) Rate used to compute the interest paid per period EAR = Effective Annual Rate the annual rate of interest actually being earned APR = Annual Percentage Rate periodic rate X the number of periods per year Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-37 How do we find EAR for a simple rate of 10%, compounded semi-annually? m i EAR = 1 + SIMPLE - 1 m 2 0.10 - 1.0 = 1+ 2 2 = 1.05 - 1.0 = 0.1025 = 10.25%. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-38 FV of $100 after 3 years under 10% semi-annual compounding? Quarterly? iSIMPLE FVn = PV 1 + m FV3s mxn 0.10 = $100 1 + 2 2x3 = $100(1.05)6 = $134.01 FV3Q = $100(1.025)12 = $134.49 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-39 Fractional Time Periods Example: $100 deposited in a bank at 10% interest for 0.75 of the year 0 0.25 0.50 0.75 1.00 8% - 100 INPUTS FV = ? 0.75 N OUTPUT 10 I/YR - 100 PV 0 PMT ? FV =107.41 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-40 Amortized Loans Amortized Loan: A loan that is repaid in equal payments over its life. Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-41 Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments. Step 1: Find the required payments 0 1 2 PMT PMT 3 10% -1000 INPUTS OUTPUT 3 10 -1000 N I/YR PV PMT 0 PMT FV 402.11 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-42 Step 2: Find interest charge for Year 1 INTt = Beg balt (i) INT1 = 1000(0.10) = $100. Step 3: Find repayment of principal in Year 1 Repmt. = PMT - INT = 402.11 - 100 = $302.11. Step 4: Find ending balance after Year 1 End bal = Beg bal - Repmt = 1000 - 302.11 = $697.89. Repeat these steps for Years 2 and 3 to complete the amortization table. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-43 Loan Amortization Table 10 Percent Interest Rate YR Beg Bal PMT 1 $1000 $402 $100 $302 $698 2 698 402 70 332 366 3 366 402 37 366 0 1,206.34 206.34 1,000 Total INT Prin PMT End Bal Interest declines. Tax Implications. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-44 $ 402.11 Interest 302.11 Principal Payments 0 1 2 3 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-45 Comparison of Different Types of Interest Rates iSIMPLE: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. iper: Used in calculations, shown on time lines. If iSIMPLE has annual compounding, then iper = iSIMPLE/1 = iSIMPLE. EAR : 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 (C) 2000 by Harcourt, Inc. All rights reserved. 6-46 Simple (Quoted) Rate iSIMPLE is stated in contracts. Periods per year (m) must also be given. Examples: 8%; Quarterly 8%, Daily interest (365 days) Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-47 Periodic Rate Periodic rate = iper = iSIMPLE/m, where m is 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 (C) 2000 by Harcourt, Inc. All rights reserved. 6-48 Effective Annual Rate Effective Annual Rate: The annual rate which cause PV to grow to the same FV as under multiperiod compounding. Example: EAR for 10%, semiannual: FV EAR = = = (1 + iSIMPLE/m)m (1.05)2 = 1.1025. 10.25% because (1.1025)1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually. Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 6-49 End of Chapter 6 The Time Value of Money Copyright (C) 2000 by Harcourt, Inc. All rights reserved.
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