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Chapter 4 The Time Value of Money 1

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Learning Outcomes Chapter 4 Identify various types of cash flow patterns Compute the future value and the present value of different cash flow streams Compute the return on an investment and how long it takes to reach a financial goal Explain the difference between the Annual Percentage Rate and the Effective Annual Rate Describe an amortized loan. Compute amortized loan payments and the amount that must be paid at a specific point during the life of the loan 2

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Time Value of Money The principles and computations used to revalue cash payoffs from different times so they are stated in dollars of the same time period. 3

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Cash Flow Time Lines 4 Graphical representations used to show timing of cash flows PV = Present Value – the beginning amount that can be invested. PV also represents the current value of some future amount. FV = Future Value – the value to which an amount invested today will grow at the end of n periods.

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Types of Cash Flow Patterns Lump Sum Amount – a single payment (received or made) that occurs either today or at some date in the future. Annuity – Multiple payments of the same amount over equal time periods. Uneven Cash Flows – Multiple payments of different amounts over a period of time. 5

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Future Value Compounding – To compute the future value of an amount we push forward the current amount by adding interest for each period in which the money can earn interest in the future. 6

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Future Value of a Lump-Sum Amount FV n 7

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Four Ways to Solve Time Value of Money Problems Use Cash Flow Time Line Use Equations Use Financial Calculator Use Electronic Spreadsheet 8

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Time Line Solution 9 The Future Value of $700 invested at 10% per year for 3 years

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10 Equation Solution FV 3 = $700(1.10) 3 = $700(1.33100) = $931.70

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Financial Calculator Solution 11

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Spreadsheet Solution - MS Excel 12 1. Set up a table that contains the data used to solve the problem 2. Click f x and choose function 3. Click the cells containing the appropriate data to calculate the answer Rate (B2) Number of periods (B1) Present value (B3) Payment (B4) (not used) Type (B5) (not used)

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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. 13

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What’s the FV of a 3-year Ordinary Annuity of $400 at 5%? 14

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Equation Solution: 15

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Financial Calculator Solution 16

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Find the FV of an Annuity Due 17

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Equation Solution 18

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Financial Calculator Solution 19

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Find the FV of an Uneven Cash Flow Stream – FVCF n 20

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Equation Solution 21

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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. 22

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Present Value of a Lump-Sum Amount PV 23

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PV of a Lump-Sum Amount Financial Calculator 24

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Present Value of an Annuity (Ordinary) 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. 25

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What is the PV of $400 due in 3 years if r = 5%? 26

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Equation Solution 27

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Financial Calculator Solution 28

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Present Value of an Annuity Due 29

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Equation Solution 30

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Financial Calculator Solution 31

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Perpetuities 32 Streams of equal payments that are expected to go on forever

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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. 33

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PV of an Uneven Cash Flow Stream – PVCF n 34

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Equation Solution 35

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Financial Calculator Solution Input in “CF” register: CF0 =0 CF1 =400 CF2 =300 CF3 =250 Enter I = 5, then press NPV button to get NPV = -869.02. 36

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Solving for Interest Rates (r) 37 You pay $78.35 for an investment that promises to pay you $100 per year for the next five years. What annual rate of return will you earn on this investment?

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Financial Calculator Solution 38

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Solving for Time (n) 39 A security costing $68.30 will provide a return of 10% per year and you want to keep the investment until it grows to a value of $100. How long will it take the investment to grow to $100?

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Financial Calculator Solution 40

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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. 41

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The FV of a lump sum be larger if we compound more often, holding the stated r constant? Why? 42 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.

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Distinguishing Between Different Interest Rates 43 Simple (Quoted) Rate r SIMPLE = Simple (Quoted) Rate used to compute the interest paid per period Effective Annual Rate r EAR = Effective Annual Rate the annual rate of interest actually being earned Annual Percentage Rate APR = Annual Percentage Rate = r SIMPLE periodic rate X the number of periods per year

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Comparison of Different Types of Interest Rates r SIMPLE : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. r PER : Used in calculations, shown on time lines. r EAR : Used to compare returns on investments with different payments per year. 44

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Simple (Quoted) Rate r SIMPLE is stated in contracts Periods per year (m) must also be given Examples: 8%, compounded quarterly 8%, compounded daily (365 days) 45

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Periodic Rate Periodic rate = r PER = r SIMPLE /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: r PER = 8/4 = 2% 8% daily (365): r PER = 8/365 = 0.021918% 46

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Effective Annual Rate 47 The annual rate that causes PV to grow to the same FV as under multi-period compounding.

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How do we find r EAR for a simple rate of 10%, compounded semi-annually? 48

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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 49

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An amortization schedule for a $15,000, 8 percent loan that requires three equal annual payments. 50

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Financial Calculator Solution 51

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Amortization Schedule 52

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