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Chapter 3 The Time Value of Money © 2005 Thomson/South-Western

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2 Time Value of Money The most important concept in finance Used in nearly every financial decision Business decisions Personal finance decisions

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3 Cash Flow Time Lines CF 0 CF 1 CF 3 CF 2 0123 k% Time 0 is today Time 1 is the end of Period 1 or the beginning of Period 2. Graphical representations used to show timing of cash flows

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4 100 012 Year k% Time line for a $100 lump sum due at the end of Year 2

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5 Time line for an ordinary annuity of $100 for 3 years 100 0123 k%

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6 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 k% -50

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7 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. Future Value

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8 Calculating FV is compounding! Question: 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? Translation: What is the FV of an initial $100 after 3 years if k = 10%? Key Formula: FV n = PV (1 + k) n

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9 Three Ways to Solve Time Value of Money Problems Use Equations Use Financial Calculator Use Electronic Spreadsheet

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10 Solve this equation by plugging in the appropriate values: Numerical (Equation) Solution PV = $100, k = 10%, and n =3

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11 Financial calculators solve this equation: There are 4 variables (FV, PV, k, n). If 3 are known, the calculator will solve for the 4th. Financial Calculator Solution

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12 First: set calculator to show 4 digits to the right of the decimal place To enter “Format” register: Type: 2 nd, period See: DEC (decimal) = (varies) Type: 4, enter See: DEC = 4 Exit Format register: hit CE/C See 0.0000 Financial Calculator Solution

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13 INPUTS OUTPUT 3 10-100 0? NI/YR PV PMTFV 133.10 Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR= 1, END Financial Calculator Solution

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14 Spreadsheet Solution Set up ProblemClick on Function Wizard and choose Financial/FV

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15 Spreadsheet Solution Reference cells: Rate = interest rate, k Nper = number of periods interest is earned Pmt = periodic payment PV = present value of the amount

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16 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 future cash flow or series of future cash flows; it is the reverse of compounding.

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17 100 0123 10% PV = ? What is the PV of $100 due in 3 years if k = 10%?

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18 INPUTS OUTPUT 3 10 ? 0100 NI/YR PV PMTFV -75.13 Financial Calculator Solution VITAL: Either PV or FV must be negative. Here PV = -75.13. Put in $75.13 today, take out $100 after 3 years.

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19 If sales grow at 20% per year, how long before sales double?

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20 INPUTS OUTPUT ?20 -1 0 2 N I/YR PV PMTFV 3.8 Graphical Illustration: 0 1234 1 2 FV 3.8 Year Financial Calculator Solution

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21 Future Value of an Annuity Annuity: A series of payments of equal amounts 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.

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22 PMT 0123 k% PMT 0123 k% PMT Ordinary Annuity Versus Annuity Due Ordinary Annuity Annuity Due

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23 100 0123 10% 110 121 FV= 331 What’s the FV of a 3-year Ordinary Annuity of $100 at 10%?

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24 Financial Calculator Solution INPUTS OUTPUT 310 0 -100 ? 331.00 NI/YRPV PMT FV

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25 Present Value of an Annuity PVA n = 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.

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26 248.69 = PV 100 0123 10% 90.91 82.64 75.13 What is the PV of this Ordinary Annuity?

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27 We know the payments but no lump sum FV, so enter 0 for future value. Financial Calculator Solution INPUTS OUTPUT 310 ? 100 0 -248.69 NI/YRPV PMT FV

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28 100 0123 10% 100 Find the FV and PV if the Annuity were an Annuity Due.

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29 ANNUITY Due: Switch from “End” to “Begin” Method: (2 nd BGN, 2 nd Enter) Then enter variables to find PVA 3 = $273.55. Then enter PV = 0 and press FV to find FV = $364.10. Financial Calculator Solution INPUTS OUTPUT 310 ? 100 0 -273.55 NI/YRPV PMT FV

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30 What is the PV of a $100 perpetuity if k = 10%? You MUST know the formula for a perpetuity: PV = PMT k So, here: PV = 100/.1 = $1000

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31 250 0123 k = ? - 846.80 4 250 You pay $846.80 for an investment that promises to pay you $250 per year for the next four years, with payments made at the end of each year. What interest rate will you earn on this investment? Solving for Interest Rates with Annuities

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32 Financial Calculator Solution INPUTS OUTPUT 4 ? -846.80 250 0 7.0 N I/YR PV PMT FV

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33 What interest rate would cause $100 to grow to $125.97 in 3 years?

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34 What interest rate would cause $100 to grow to $125.97 in 3 years? INPUTS OUTPUT 3 ? -100 0 125.97 8% NI/YRPV PMT FV

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35 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—that is, an annuity stream. Cash flow (CF) designates cash flows in general, both constant cash flows and uneven cash flows.

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36 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV What is the PV of this Uneven Cash Flow Stream?

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37 Financial Calculator Solution In “CF” register, input the following: CF0 =0 C01 =100F01 =1 C02 =300F01 =1 C03 =300F01 =1 C04 =-50F01 =1 In “NPV” Register: Enter I = 10% Hit down arrow to see “NPV = 0” Hit CPT for compute See “NPV = 530.09” (Here NPV = PV.)

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38 Semiannual and Other Compounding Periods Annual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added once a year. Semiannual compounding is the process of determining the future value of a cash flow or series of cash flows when interest is added twice a year.

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39 Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated k constant? Why?

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40 If compounding is more frequent than once a year—for example, semi-annually, quarterly, or daily—interest is earned on interest—that is, compounded—more often. Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated k constant? Why? LARGER!

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41 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. Compounding Annually vs. Semi-Annually

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42 Simple (Quoted) Rate k SIMPLE = Simple (Quoted) Rate Periodic Rate k PER = Periodic Rate Effective Annual Rate Annual Percentage Rate EAR= Effective Annual Rate APR = Annual Percentage Rate Distinguishing Between Different Interest Rates

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43 Simple (Quoted) Rate k SIMPLE = Simple (Quoted) Rate *used to compute the interest paid per period *stated in contracts, quoted by banks & brokers *number of periods per year must also be given *Not used in calculations or shown on time lines Examples: 8%, compounded quarterly 8%, compounded daily (365 days) k SIMPLE

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44 Periodic Rate = k Per k PER : Used in calculations, shown on time lines. If k SIMPLE has annual compounding, then k PER = k SIMPLE k PER = k SIMPLE /m, where m is number of compounding periods per year. Determining m: m = 4 for quarterly m = 12 for monthly m = 360 or 365 for daily compounding Examples: 8% quarterly: k PER = 8/4 = 2% 8% daily (365): k PER = 8/365 = 0.021918%

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45 Annual Percentage Rate APR = Annual Percentage Rate = k SIMPLE periodic rate X the number of periods per year APR = k simple

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46 Effective Annual Rate EAR= Effective Annual Rate * the annual rate of interest actually being earned * The annual rate that causes PV to grow to the same FV as under multi-period compounding. * Use to compare returns on investments with different payments per year. * Use for calculations when dealing with annuities where payments don’t match interest compounding periods. EAR

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47 How to find EAR for a simple rate of 10%, compounded semi-annually Hit 2 nd then 2 to enter “ICONV” register: NOM = simple interest rate Type: 10, enter, down arrow twice C/Y = compounding periods per year Type: 2, enter, up arrow (or down arrow twice) EFF = effective annual rate = EAR Type: CPT (for compute) See: 10.25 POINT: Any PV would grow to same FV at 10.25% annually or 10% semiannually.

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48 Continuous Compounding The formula is FV = PV(e kt ) k = the interest rate (expressed as a decimal) t = number of years Calculator “workaround” Store 9999999999 (as many nines as possible) in your calculator under STO + 9 Then, for N: N = # of years times “RCL 9” Then, for I: I = simple interest divided by “RCL 9”

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49 Continuous Compounding Question: What is the value of a $1,000 deposit invested for 5 years at an interest rate of 10%, compounded continuously? INPUTS OUTPUT 5* 10/ RCL9 RCL9 -1000 0 ? 1648.72 N I/YR PV PMT FV

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50 Fractional Time Periods 00.250.500.75 10% - 100 1.00 FV = ? What is the value of $100 deposited in a bank at EAR = 10% for 0.75 of the year?

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51 Fractional Time Periods 00.250.500.75 10% - 100 1.00 FV = ? What is the value of $100 deposited in a bank at EAR = 10% for 0.75 of the year? INPUTS OUTPUT 0.75 10 -100 0 ? 107.41 N I/YR PV PMT FV

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52 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, and so forth to determine how much of each payment represents principal repayment and how much represents interest. They are very important, especially to homeowners! Financial calculators (and spreadsheets) are great for setting up amortization tables.

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53 Task: Construct an amortization schedule for a $1,000, 10 percent loan that requires three equal annual payments. PMT 0123 10% -1,000

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54 PMT 0123 10% -1000 INPUTS OUTPUT 3 10 -1000 ? 0 402.11 NI/YRPV PMT FV Step 1: Determine the required payments

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55 Hit 2 nd, PV to enter “Amort Register” For 1 st principal payment, 1 st interest payment, and 1 st year remaining balance, enter: P1 = 1 P2 = 1 Down arrow See: Bal = -697.88, down arrow PRN = 302.11 INT = 100 Enter “Amort” Register

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56 Interest declines, which has tax implications. Step 2: Create Loan Amortization Table * Rounding difference

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57 Hit 2 nd, PV to enter “Amort Register” For 2nd principal payment, 2nd interest payment, and 2 nd year remaining balance, enter: P1 = 2 P2 = 2 Down arrow See: Bal = -365.56, down arrow PRN = 332.32 INT = 69.79 Enter “Amort” Register

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58 Interest declines, which has tax implications. Step 2: Create Loan Amortization Table * Rounding difference

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59 Hit 2 nd, PV to enter “Amort Register” For 3 rd principal payment, 3rd interest payment, and 3rd year remaining balance, enter: P1 = 3 P2 = 3 Down arrow See: Bal = 0, down arrow PRN = 365.56 INT = 36.65 Enter “Amort” Register

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60 Interest declines, which has tax implications. Step 2: Create Loan Amortization Table * Rounding difference

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61 1. Review Chapter 3 materials 2.Do Chapter 3 homework 3.Prepare for Chapter 3 quiz 3. Read Chapter 4 Before Next Class

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