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Chapter 3 The Time Value of Money

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

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

  3. 0 1 2 3 k% CF0 CF1 CF2 CF3 Cash Flow Time Lines Graphical representations used to show timing of cash flows Time 0 is todayTime 1 is the end of Period 1 or the beginning of Period 2.

  4. 0 1 2 Year k% 100 Time line for a $100 lump sumdue at the end of Year 2

  5. 0 1 2 3 k% 100 100 100 Time line for an ordinary annuity of $100 for 3 years

  6. 0 1 2 3 k% 100 75 50 -50 Time line for uneven CFs - $50 at t = 0 and $100, $75, and $50 at the end of Years 1 through 3

  7. 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.

  8. Future Value 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? FVn = FV1 = PV + INT = PV + (PV x k) = PV (1 + k) = $100(1 + 0.05) = $100(1.05) = $105

  9. 0 1 2 3 10% 100 FV = ? Finding FV is Compounding. What’s the FV of an initial $100 after 3 years if k = 10%?

  10. Future Value After 1 year: FV1 = PV + Interest1 = PV + PV (k) = PV(1 + k) = $100 (1.10) = $110.00. After 2 years: FV2 = PV(1 + k)2 = $100 (1.10)2 = $121.00. After 3 years: FV3 = PV(1 + k)3 = 100 (1.10)3 = $133.10. In general, FVn = PV (1 + k)n

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

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

  13. Present Value • Present value is the value today of a future cash flow or series of cash flows. • Discountingis the process of finding the present value of a future cash flow or series of future cash flows; it is the reverse of compounding.

  14. 0 1 2 3 10% PV = ? 100 What is the PV of $100 due in 3 years if k = 10%?

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

  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.

  17. 0 1 2 3 k% PMT PMT PMT 0 1 2 3 k% PMT PMT PMT Ordinary Annuity Versus Annuity Due Ordinary Annuity Annuity Due

  18. 0 1 2 3 10% 100 100 100 110 121 What’s the FV of a 3-year Ordinary Annuity of $100 at 10%? FV = 331

  19. Numerical Solution (using table):

  20. 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.

  21. 0 1 2 3 10% 100 100 100 90.91 82.64 75.13 What is the PV of this Ordinary Annuity? 248.69 = PV

  22. Using table: © 2005 Thomson/South-Western

  23. 0 1 2 3 10% 100 100 100 Find the FV and PV if theAnnuity were an Annuity Due.

  24. Numerical Solution

  25. 0 1 2 3 4 k = ? - 864.80 250 250 250 250 Solving for Interest Rateswith annuities 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?

  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.

  27. 1 2 3 100 300 300 4 0 10% -50 90.91 247.93 225.39 -34.15 530.08 = PV What is the PV of this Uneven Cash Flow Stream?

  28. Numerical Solution

  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.

  30. 0 1 2 3 10% 100 133.10 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 Compounding Annually vs. Semi-Annually Annually: FV3 = 100(1.10)3 = 133.10. Semi-annually: FV6/2 = 100(1.05)6 = 134.01.

  31. Distinguishing Between Different Interest Rates kSIMPLE = Simple (Quoted) Rateused to compute the interest paid per period EAR = Effective Annual Ratethe annual rate of interest actually being earned APR = Annual Percentage Rate= kSIMPLEperiodic rate X the number of periods per year

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

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

  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

  35. 0 1 2 3 10% PMT PMT PMT -1,000 Step 1: Construct an amortization schedule for a $1,000, 10% loan that requires 3 equal annual payments.

  36. Step 2: Find interest charge for Year 1 INTt = Beginning balancet x (k)INT1 = 1,000 x 0.10 = $100.00 Step 3: Find repayment of principal in Year 1 Repayment = PMT - INT = $402.11 - $100.00 = $302.11.

  37. Step 4: Find ending balance after Year 1 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.

  38. Loan Amortization Table10% Interest Rate * Rounding difference Interest declines, which has tax implications.