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Chapter 3 The Time Value of Money

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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 FV n = FV 1 = PV + INT = PV + (PV x k) = PV (1 + k) = $100(1 + 0.05) = $100(1.05) = $105 How much would you have at the end of one year if you deposited $100 in a bank account that pays 5% interest each year? Future Value

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9 FV = ? 0123 10% 100 Finding FV is Compounding. What’s the FV of an initial $100 after 3 years if k = 10%?

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10 After 1 year: FV 1 =PV + Interest 1 = PV + PV (k) =PV(1 + k) =$100 (1.10) =$110.00. After 2 years: FV 2 =PV(1 + k) 2 =$100 (1.10) 2 =$121.00. After 3 years: FV 3 =PV(1 + k) 3 =100 (1.10) 3 =$133.10. In general, FV n = PV (1 + k) n Future Value

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

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

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13 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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14 100 0123 10% PV = ? What is the PV of $100 due in 3 years if k = 10%?

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15 Solve FV n = PV (1 + k ) n for PV:

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16 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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17 PMT 0123 k% PMT 0123 k% PMT Ordinary Annuity Versus Annuity Due Ordinary Annuity Annuity Due

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18 100 0123 10% 110 121 FV= 331 What’s the FV of a 3-year Ordinary Annuity of $100 at 10%?

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19 Numerical Solution (using table):

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20 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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21 248.69 = PV 100 0123 10% 90.91 82.64 75.13 What is the PV of this Ordinary Annuity?

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22 Using table:

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23 100 0123 10% 100 Find the FV and PV if the Annuity were an Annuity Due.

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24 Numerical Solution

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25 250 0123 k = ? - 864.80 4 250 You pay $864.80 for an investment that promises to pay you $250 per year for the next 4 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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26 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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27 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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28 Numerical Solution

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29 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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30 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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31 Simple (Quoted) Rate k 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 = k SIMPLE periodic rate X the number of periods per year Distinguishing Between Different Interest Rates

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

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33 FV of $100 after 3 years if interest is 10% compounded semi-annual? Quarterly?

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34 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

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35 Step 1: Construct an amortization schedule for a $1,000, 10% loan that requires 3 equal annual payments. PMT 0123 10% -1,000

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36 Step 2: Find interest charge for Year 1 INT t = Beginning balance t x (k) INT 1 = 1,000 x 0.10 = $100.00 Repayment= PMT - INT = $402.11 - $100.00 = $302.11. Step 3: Find repayment of principal in Year 1

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37 End bal=Beginning bal. - Repayment =$1,000 - $302.11 = $697.89. Repeat these steps for the remainder of the payments (Years 2 and 3 in this case) to complete the amortization table. Step 4: Find ending balance after Year 1

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38 Interest declines, which has tax implications. Loan Amortization Table 10% Interest Rate * Rounding difference

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