1. Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 9 The Time Value of Money 1

2. Chapter 9- Learning Objectives Identify various types of cash flow patterns (streams) that are observed in business. Compute (a) the future values and (b) the present values of different cash flow streams, and explain the results. Compute (a) the return (interest rate) on an investment (loan) and (b) how long it takes to reach a financial goal. Explain the difference between the Annual Percentage Rate (APR) and the Effective Annual Rate (EAR), and explain when each is more appropriate to use. Describe an amortized loan, and compute (a) amortized loan payments and (b) the balance (amount owed) on an amortized loan at a specific point during its life. Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2

3. Time Value of Money The principles and computations used to revalue cash payoffs at different times so they are stated in dollars of the same time period The most important concept in finance used in nearly every financial decision Business decisions Personal finance decisions Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 3

4. Cash Flow Patterns Lump-sum amount – a single payment paid or received in the current period or some future period Annuity - A series of equal payments that occur at equal time intervals Uneven cash flow stream – multiple payments that are not equal and do not occur at equal intervals Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 4

5. Cash Flow Time Lines Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 5 01 23 6% Time: PV = 100Cash Flows: FV = ? Time 0 is today Time 1 is the end of Period 1 or the beginning of Period 2, and so forth. Graphical representations used to show timing of cash flows

6. 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. Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 6

7. Future Value How much would you have at the end of one year if you deposit $100 in a bank account that pays 5 percent interest each year? 7 FV n = FV 1 = PV + INT = PV + PV(r) = PV (1 + r) = $100(1 + 0.05) = $100(1.05) = $105 Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

8. What’s the FV of an initial $700 after three years if r = 10%? Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 8 FV = ? 0123 10% 700 Finding FV is Compounding

9. Future Value Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 9 After 1 year: FV 1 =PV + Interest 1 = PV + PV (r) =PV(1 + r) =$700 (1.10) =$770.00. After 2 years: FV 2 =PV(1 + r) 2 =$700 (1.10) 2 =$847.00. After 3 years: FV 3 =PV(1 + r) 3 =$700 (1.10) 3 =$931.70 In general, FV n = PV (1 + r) n

10. Three Ways to Solve Time Value of Money Problems Use Equations Use Financial Calculator Use Electronic Spreadsheet Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 10

11. Numerical (Equation) Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 11 PV = $700, r = 10%, and n =3

12. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 12 There are 4 variables. If 3 are known, the calculator will solve for the 4th.

13. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 13 INPUTS OUTPUT 3 10 -700 0? N I/Y PV PMTFV =931.70 Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR= 1, END

14. Spreadsheet Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 14 According to the equation shown in cell C8, the input values must be entered in a specific order: I/Y, N, PMT, PV, and PMT type (not used for this problem).

15. If sales grow at 20% per year, how long before sales double? Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 15 Solve for n: FV n = 1(1 + r) n 2 = 1(1.20) n The numerical solution is somewhat difficult.

16. Graphical Illustration Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 16 01234 1 2 3.8 Year FV

17. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 17 INPUTS OUTPUT ?20 -1 0 2 N I/Y PV PMTFV 3.8

18. Future Value of an Annuity Annuity: A series of payments of equal amounts at equal 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. Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 18

19. Ordinary Annuity Versus Annuity Due Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 19 PMT 0123 r% PMT 0 123 r% PMT Ordinary Annuity Annuity Due

20. FV of a 3-year Ordinary Annuity of $400 at 5% Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 20

21. Numerical Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 21

22. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 22 INPUTS OUTPUT 3 5 0 -400 ? =1261.00 NI/YPV PMT FV

23. Spreadsheet Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 23

24. FV of a 3-year Annuity Due of $400 at 5% Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 24

25. Numerical Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 25

26. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 26 INPUTS OUTPUT 3 5 0 -400 ? =1,324.05 NI/YPV PMT FV BGN

27. Spreadsheet Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 27 N =3 I/Y =5.00% PV =$0.00 PMT =-$400.00 PMT type =1(0 = ordinary annuity; 1 = annuity due) FV =? The equation used to solve for FV in cell B8 Values that correspond to the cell reference in cell C8 FVA(DUE) =$1,324.05=FV(B2,B1,B4,B3,B5=FV(0.05,3,-400,0,1)

28. Future Value of an Uneven Cash Flow Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 28

29. Numerical Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 29

30. 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. Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 30

31. What is the PV of $935 due in three years if r = 10%? Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 31

32. Numerical Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 32 Solve FV n = PV (1 + r ) n for PV:

33. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 33 INPUTS OUTPUT 3 10 ? 0935 N I/Y PV PMTFV -702.48 Either PV or FV must be negative. Here PV = -702.48. Invest $702.48 today, take out $935 after 3 years.

34. Spreadsheet Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 34

35. 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 Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 35

36. What is the PV of this Ordinary Annuity? Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 36

37. Numerical Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 37

38. Financial Calculator Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 38 We know the payments but there is no lump sum FV, so enter 0 for future value. INPUTS OUTPUT 3 5 ? 400 0 = -1,089.30 NI/YPV PMT FV

39. Spreadsheet Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 39

40. Present Value of an Annuity Due Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 40 1,143.76 = PVA (DUE) 3 400 0123 05% 400.00 380.95 362.81 (1.05) x Value of Each Deposit Today (Year 0)

41. Numerical Solution Principles of Finance 5e, 9 The Time Value of Money © 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 41