The Time Value of Money

1. The Time Value of Money References: Keown, 2005, Financial Management: Principles and Applications, 10th ed., Prentice Hall Ross, Westerfield, and Jordan, 2006, Fundamentals of Corporate Finance, 7th ed., McGraw-Hill Brigham & Houston, Fundamentals of Financial Management, 8th ed. Prepared By:Liem Pei Fun, S.E., MCom.

2. About me • 1998-2002 Bachelor of Economics (Majoring in Finance), Petra Christian University • 2004-2005 Postgraduate Diploma in Finance, The University of Melbourne • 2005-2006 Master of Commerce (Finance), The University of Melbourne • Pass CFA Exam Level I • Bloomberg Global Product Certification – Equity Level One • Head of Finance Program, Faculty of Economics, Petra Christian University

3. OUTLINE • Future Value Single Sum • Present Value Single Sum • Future Value Annuities • Present Value Annuities • Ordinary Annuity vs Annuity Due • Perpetuities • Practice Problems

4. The Time Value of Money  2002, Prentice Hall, Inc.

5. Compounding and Discounting Single Sums

6. Today Future We know that receiving $1 today is worth more than $1 in the future. This is duetoopportunity costs. The opportunity cost of receiving $1 in the future is theinterestwe could have earned if we had received the $1 sooner.

7. Today Future ? Today Future ? If we can measure this opportunity cost, we can: • Translate $1 today into its equivalent in the future(compounding). • Translate $1 in the future into its equivalent today(discounting).

8. Future Value

9. PV = -100 FV = 106 0 1 Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i)n FV = 100 (1.06)1 = $106

10. Calculator Keys • Texas Instruments BA-II Plus • FV = future value • PV = present value • I/Y = period interest rate • P/Y must equal 1 for the I/Y to be the period rate • Interest is entered as a percent, not a decimal • N = number of periods • Remember to clear the registers (CLR TVM) after each problem • Other calculators are similar in format

11. Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 1 year? Calculator Solution: P/Y = 1 I/Y = 6 N = 1 PV = -100 FV = $106 PV = -100 FV = 106 0 1

12. PV = -100 FV = 133.82 0 5 Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i)n FV = 100 (1.06)5 = $133.82

13. Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? Calculator Solution: P/Y = 1 I/Y = 6 N = 5 PV = -100 FV = $133.82 PV = -100 FV = 133.82 0 5

14. PV = -100 FV = 134.68 0 20 Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i/m) m x n FV = 100 (1.015)20 = $134.68

15. PV = -100 FV = 134.68 0 20 Future Value - single sumsIf you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years? Calculator Solution: P/Y = 4 I/Y = 6 N = 20 PV = -100 FV = $134.68

16. PV = -100 FV = 134.89 0 60 Future Value - single sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? Mathematical Solution: (Arithmetic Method) FV = PV (1 + i/m) m x n FV = 100 (1.005)60 = $134.89

17. PV = -100 FV = 134.89 0 60 Future Value - single sumsIf you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years? Calculator Solution: P/Y = 12 I/Y = 6 N = 60 PV = -100 FV = $134.89

18. PV = -1000 FV = $2.98m 0 100 Future Value - continuous compoundingWhat is the FV of $1,000 earning 8% with continuous compounding, after 100 years? Mathematical Solution: (Arithmetic Method) FV = PV (e in) FV = 1000 (e .08x100) = 1000 (e 8) FV = $2,980,957.99

19. Present Value

20. PV = -94.34 FV = 100 0 1 Present Value - single sumsIf you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? Mathematical Solution: (Arithmetic Method) PV = FV / (1 + i)n PV = 100 / (1.06)1 = $94.34

21. PV = -94.34 FV = 100 0 1 Present Value - single sumsIf you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%? Calculator Solution: P/Y = 1 I/Y = 6 N = 1 FV = 100 PV = -94.34

22. PV = -74.73 FV = 100 0 5 Present Value - single sumsIf you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? Mathematical Solution: (Arithmetic Method) PV = FV / (1 + i)n PV = 100 / (1.06)5 = $74.73

23. PV = -74.73 FV = 100 0 5 Present Value - single sumsIf you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%? Calculator Solution: P/Y = 1 I/Y = 6 N = 5 FV = 100 PV = -74.73

24. Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? Mathematical Solution: PV = FV / (1 + i)n 5,000 = 11,933 / (1+ i)5 .419 = ((1/ (1+i)5) 2.3866 = (1+i)5 (2.3866)1/5 = (1+i) i = .19

25. PV = -5000 FV = 11,933 0 5 Present Value - single sumsIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return? Calculator Solution: P/Y = 1 N = 5 PV = -5,000 FV = 11,933 I/Y = 19%

26. Present Value - single sumsSuppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Mathematical Solution: PV = FV / (1 + i)N 100 = 500 / (1+ .008)N 5 = (1.008)N ln 5 = ln (1.008)N ln 5 = N ln (1.008) 1.60944 = .007968 N N = 202 months

27. PV = -100 FV = 500 0 ? Present Value - single sumsSuppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500? Calculator Solution: • P/Y = 12 FV = 500 • I/Y = 9.6 PV = -100 • N = 202 months

28. 0 1 2 3 4 Compounding and Discounting Cash Flow Streams

29. Ordinary Annuity 0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT What is the difference between an ordinary annuity and an annuity due?

30. 0 1 2 3 4 Ordinary Annuity • a sequence of equal cash flows, occurring at the end of each period.

31. Examples of Annuities: • If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. • If you borrow money to buy a house or a car, you will pay a stream of equal payments.

32. 0 1 2 3 Future Value - annuityIf you invest $1,000 each year at 8%, how much would you have after 3 years?

33. Future Value - annuityIf you invest $1,000 each year at 8%, how much would you have after 3 years? Mathematical Solution: = $3,246.40

34. 0 1 2 3 Future Value - annuityIf you invest $1,000 each year at 8%, how much would you have after 3 years? Calculator Solution: P/Y = 1 I/Y = 8 PMT = -1,000 N = 3 FV = $3,246.40 1000 1000 1000

35. 0 1 2 3 Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

36. Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Mathematical Solution:

37. 0 1 2 3 Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%? Calculator Solution: P/Y = 1 I/Y = 8 PMT = -1,000 N = 3 PV = $2,577.10 1000 1000 1000

38. $1000 $1000 $1000 4 5 6 7 8 Ordinary Annuity vs. Annuity Due

39. 1000 1000 1000 4 5 6 7 8 Begin Mode vs. End Mode

40. 1000 1000 1000 4 5 6 7 8 Begin Mode vs. End Mode year year year 5 6 7 ordinary annuity

41. 1000 1000 1000 4 5 6 7 8 PV in END Mode Begin Mode vs. End Mode year year year 5 6 7 ordinary annuity

42. 1000 1000 1000 4 5 6 7 8 PV in END Mode FV in END Mode Begin Mode vs. End Mode year year year 5 6 7 ordinary annuity

43. 1000 1000 1000 4 5 6 7 8 Begin Mode vs. End Mode year year year 6 7 8 annuity due

44. 1000 1000 1000 4 5 6 7 8 PV in BEGIN Mode Begin Mode vs. End Mode year year year 6 7 8 annuity due

45. 1000 1000 1000 4 5 6 7 8 PV in BEGIN Mode FV in BEGIN Mode Begin Mode vs. End Mode year year year 6 7 8 annuity due

46. 1000 1000 1000 0 1 2 3 Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: • The Future Value (at 3) is $3,246.40. • The Present Value (at 0) is $2,577.10.

47. 0 1 2 3 What about this annuity? 1000 1000 1000 • Same 3-year time line, • Same 3 $1000 cash flows, but • The cash flows occur at the beginning of each year, rather than at the end of each year. • This is an “annuity due.”

48. 0 1 2 3 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

49. Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution:Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n) (1 + i) FV = 1,000 (FVIFA .08, 3) (1.08) use FVIFA table or = $3,506.11

50. 0 1 2 3 Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?