# 6-1 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Chapter 6 The Time Value of Money Future Value Present Value Rates of Return Amortization.

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6-1 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 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 32887- 6777.

6-2 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Cash Flow Time Lines CF 0 CF 1 CF 3 CF 2 0123 i% Tick marks 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. Graphical representations used to show the timing of cash flows.

6-3 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Time line for a \$100 lump sum due at the end of Year 2. 100 012 Year i%

6-4 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Time line for an ordinary annuity of \$100 for 3 years. 100 0123 i%

6-5 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Time line for uneven CFs -\$50 at t=0 and \$100, \$75, and \$50 at the end of Years 1 through 3. 100 50 75 0123 i% -50

6-6 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 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? FV n = FV 1 = PV + INT = PV + (PV x i) = PV (1 + i) = \$100(1+0.05) = \$100(1.05) = \$105

6-7 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Finding FVs is Compounding. What’s the FV of an initial \$100 after 3 years if i = 10%? FV = ? 0123 i=10% 100

6-8 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. In general, FV n = PV (1 + i) n After 3 years: FV 3 =PV(1 + i) 3 =100 (1.10) 3 =\$133.10. After 2 years: FV 2 =PV(1 + i) 2 =\$100 (1.10) 2 =\$121.00. After 1 year: FV 1 =PV + i = PV + PV (i) =PV(1 + i) =\$100 (1.10) =\$110.00. Future Value

6-9 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 4 Solve the Equation with a Regular Calculator 4 Use Tables 4 Use a Financial Calculator Three ways to find Future Values:

6-10 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Financial Calculator Solution: Financial calculators solve this equation: There are 4 variables. If 3 are known, the calculator will solve for the 4 th.

6-11 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 310-100 0 NI/YR PV PMTFV 133.10 INPUTS OUTPUT Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR= 1 Financial Calculator Solution:

6-12 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Present Value l Present value l Present value is the value today of a future cash flow or series of cash flows. l Discounting l Discounting is the process of finding the present value of a cash flow or series of cash flows, the reverse of compounding.

6-13 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. What is the PV of \$100 due in 3 years if i = 10%? 100 0123 10% PV = ?

6-14 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Solve FV n = PV (1 + i ) n for PV: What is the PV of \$100 due in 3 years if i = 10%?

6-15 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Either PV or FV must be negative. Here PV = -75.13. Put in \$75.13 today, take out \$100 after 3 years. 310 0100 NI/YR PV PMTFV -75.13 INPUTS OUTPUT Financial Calculator Solution:

6-16 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. If sales grow at 20% per year, how long before sales double? Solve for n: FV n = 1(1 + i) n ; 2 = 1(1.20) n.

6-17 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 20 -1 0 2 NI/YR PV PMTFV 3.8 INPUTS OUTPUT 1234 Year 0 1 2 FV 3.8 Graphical Illustration: Financial Calculator Solution:

6-18 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Future Value of an Annuity èAnnuity: èAnnuity: A series of payments of an equal amount at fixed intervals for a specified number of periods. èOrdinary (deferred) Annuity: èOrdinary (deferred) Annuity: An annuity whose payments occur at the end of each period. èAnnuity Due: èAnnuity Due: An annuity whose payments occur at the beginning of each period.

6-20 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. What’s the FV of a 3-year Ordinary Annuity of \$100 at 10%? 100 0123 10% 110 121 FV= 331

6-22 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Present Value of an Annuity èPVA n = the present value of an annuity with n payments èEach payment is discounted, and the sum of the discounted payments is the present value of the annuity

6-23 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. What is the PV of this Ordinary Annuity? 248.69 = PV 100 0123 10% 90.91 82.64 75.13

6-24 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 310 100 0 -248.69 Have payments but no lump sum FV, so enter 0 for future value. Financial Calculator Solution: NI/YRPVPMTFV INPUTS OUTPUT

6-25 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Find the FV and PV if the Annuity were an Annuity Due. 100 0123 10% 100

6-26 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 310 100 0 -273.55 INPUTS OUTPUT NI/YRPVPMTFV Switch from “End” to “Begin”. Then enter variables to find PVA 3 = \$273.55. Then enter PV = 0 and press FV to find FV = \$364.10. Financial Calculator Solution:

6-27 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Solving for Interest Rates with Annuities 250 0123 i = ? - 864.80 4 250 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?

6-28 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Financial Calculator Solution: PVA n = PMT(PVIFA i,n ) \$846.80= \$250(PVIFA i = ?,4 ) 4 ? - 846.80 250 0 =7.0 INPUTS OUTPUT NI/YRPVPMTFV INPUTS OUTPUT NI/YRPVPMTFV

6-29 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Uneven Cash Flow Streams èA series of cash flows in which the amount varies from one period to the next. Ù Payment (PMT) designates constant cash flows Ù Cash flow (CF) designates cash flows in general, including uneven cash flows

6-30 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. What is the PV of this Uneven Cash Flow Stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

6-31 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 4 Input in “Cash Flow” register: CF 0 = 0 CF 1 =100 CF 2 =300 CF 3 =300 CF 4 = -50 4 Enter I = 10%, then press NPV button to get NPV = 530.09. (Here NPV = PV.) Financial Calculator Solution:

6-32 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. \$100 (1 + i ) 3 = \$125.97. What interest rate would cause \$100 to grow to \$125.97 in 3 years? 3-100 0 125.97 INPUTS OUTPUT NI/YRPVFV PMT 8%

6-33 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Semiannual and Other Compounding Periods èAnnual compounding èAnnual 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. èSemiannual compounding èSemiannual 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.

6-34 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. LARGER! If compounding is more frequent than once a year--for example, semi-annually, quarterly, or daily-- interest is earned on interest more often. Will the FV of a lump sum be larger or smaller if we compound more often, holding the state i% constant? Why?

6-35 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 0123 10% 100 133.10 0123 5% 456 134.01 123 0 100 Annually: FV 3 = 100(1.10) 3 = 133.10. Semi-annually: FV 6/2 = 100(1.05) 6 = 134.01.

6-36 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Distinguishing Between Different Interest Rates Simple (Quoted) Rate i SIMPLE = Simple (Quoted) Rate used to compute the interest paid per period Effective Annual Rate EAR= Effective Annual Rate the annual rate of interest actually being earned Annual Percentage Rate APR = Annual Percentage Rate periodic rate X the number of periods per year

6-37 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. How do we find EAR for a simple rate of 10%, compounded semi-annually? m EAR= 1+ i SIMPLE     m - 1   = 1+ 0.10 2 - 1.0 = 1.05 - 1.0 = 0.1025= 10.25%.       2 2

6-38 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. FV of \$100 after 3 years under 10% semi-annual compounding? Quarterly? = \$100(1.05) 6 = \$134.01 FV 3Q = \$100(1.025) 12 = \$134.49

6-39 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 0.75 10 - 100 0 ? =107.41 INPUTS OUTPUT NI/YRPVPMTFV Fractional Time Periods 00.250.500.75 8% - 100 1.00 FV = ? Example: \$100 deposited in a bank at 10% interest for 0.75 of the year

6-40 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Amortized Loans l Amortized Loan: l Amortized Loan: A loan that is repaid in equal payments over its life. l Amortization tables are widely used-- for home mortgages, auto loans, business loans, retirement plans, etc. lThey are very important! l Financial calculators (and spreadsheets) are great for setting up amortization tables.

6-41 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Construct an amortization schedule for a \$1,000, 10% annual rate loan with 3 equal payments. Step 1: Find the required payments PMT 0123 10% -1000 3 10 -1000 0 INPUTS OUTPUT NI/YRPVFV PMT 402.11

6-42 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. Step 2: Find interest charge for Year 1 Step 2: Find interest charge for Year 1 INT t = Beg bal t (i) INT 1 = 1000(0.10) = \$100. Step 3: Find repayment of principal in Year 1 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 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.

6-44 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. \$ 0123 402.11 Interest Principal Payments 302.11 Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling.

6-45 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 4 i SIMPLE : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. 4 i per : Used in calculations, shown on time lines. If i SIMPLE has annual compounding, then i per = i SIMPLE /1 = i SIMPLE. 4 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.) Comparison of Different Types of Interest Rates

6-46 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 4 i SIMPLE is stated in contracts. Periods per year (m) must also be given. 4 Examples: 8%; Quarterly 8%, Daily interest (365 days) Simple (Quoted) Rate

6-47 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 4 Periodic rate = i per = i SIMPLE /m, where m is periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. 4 Examples: 8% quarterly: i per = 8/4 = 2% 8% daily (365): i per = 8/365 = 0.021918% Periodic Rate

6-48 Copyright (C) 2000 by Harcourt, Inc. All rights reserved. 4 Effective Annual Rate: The annual rate which cause PV to grow to the same FV as under multiperiod compounding. 4 Example: EAR for 10%, semiannual: FV=(1 + i SIMPLE /m) m =(1.05) 2 = 1.1025. EAR=10.25% because (1.1025) 1 = 1.1025. Any PV would grow to same FV at 10.25% annually or 10% semiannually. Effective Annual Rate