2. © The McGraw-Hill Companies, Inc.,2001 3- 1 Irwin/McGraw-Hill Chapter 3 Fundamentals of Corporate Finance Third Edition The Time Value of Money Brealey Myers Marcus slides by Matthew Will Irwin/McGraw-Hill © The McGraw-Hill Companies, Inc.,2001

3. 3- 2 Irwin/McGraw-Hill Topics Covered  Future Values  Present Values  Multiple Cash Flows  Perpetuities and Annuities  Inflation & Time Value  Effective Annual Interest Rate

4. © The McGraw-Hill Companies, Inc.,2001 3- 3 Irwin/McGraw-Hill Future Values Future Value - Amount to which an investment will grow after earning interest. Compound Interest - Interest earned on interest. Simple Interest - Interest earned only on the original investment.

5. © The McGraw-Hill Companies, Inc.,2001 3- 4 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100.

6. © The McGraw-Hill Companies, Inc.,2001 3- 5 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. Interest Earned Per Year = 100 x.06 = $ 6

7. © The McGraw-Hill Companies, Inc.,2001 3- 6 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100.

8. © The McGraw-Hill Companies, Inc.,2001 3- 7 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. TodayFuture Years 1 2 3 4 5 Interest Earned Value100

9. © The McGraw-Hill Companies, Inc.,2001 3- 8 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. TodayFuture Years 1 2 3 4 5 Interest Earned 6 Value100106

10. © The McGraw-Hill Companies, Inc.,2001 3- 9 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. TodayFuture Years 1 2 3 4 5 Interest Earned 6 6 Value100106112

11. © The McGraw-Hill Companies, Inc.,2001 3- 10 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. TodayFuture Years 1 2 3 4 5 Interest Earned 6 6 6 Value100106112118

12. © The McGraw-Hill Companies, Inc.,2001 3- 11 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. TodayFuture Years 1 2 3 4 5 Interest Earned 6 6 6 6 Value100106112118124

13. © The McGraw-Hill Companies, Inc.,2001 3- 12 Irwin/McGraw-Hill Future Values Example - Simple Interest Interest earned at a rate of 6% for five years on a principal balance of $100. TodayFuture Years 1 2 3 4 5 Interest Earned 6 6 6 6 6 Value100106112118124130 Value at the end of Year 5 = $130

14. © The McGraw-Hill Companies, Inc.,2001 3- 13 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance.

15. © The McGraw-Hill Companies, Inc.,2001 3- 14 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. Interest Earned Per Year =Prior Year Balance x.06

16. © The McGraw-Hill Companies, Inc.,2001 3- 15 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. TodayFuture Years 1 2 3 4 5 Interest Earned Value100

17. © The McGraw-Hill Companies, Inc.,2001 3- 16 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. TodayFuture Years 1 2 3 4 5 Interest Earned 6.00 Value100106.00

18. © The McGraw-Hill Companies, Inc.,2001 3- 17 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. TodayFuture Years 1 2 3 4 5 Interest Earned 6.00 6.36 Value100106.00112.36

19. © The McGraw-Hill Companies, Inc.,2001 3- 18 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. TodayFuture Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 Value100106.00112.36119.10

20. © The McGraw-Hill Companies, Inc.,2001 3- 19 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. TodayFuture Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 7.15 Value100106.00112.36119.10126.25

21. © The McGraw-Hill Companies, Inc.,2001 3- 20 Irwin/McGraw-Hill Future Values Example - Compound Interest Interest earned at a rate of 6% for five years on the previous year’s balance. TodayFuture Years 1 2 3 4 5 Interest Earned 6.00 6.36 6.74 7.15 7.57 Value100106.00112.36119.10126.25133.82 Value at the end of Year 5 = $133.82

22. © The McGraw-Hill Companies, Inc.,2001 3- 21 Irwin/McGraw-Hill Future Values Future Value of $100 = FV

23. © The McGraw-Hill Companies, Inc.,2001 3- 22 Irwin/McGraw-Hill Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

24. © The McGraw-Hill Companies, Inc.,2001 3- 23 Irwin/McGraw-Hill Future Values Example - FV What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

25. © The McGraw-Hill Companies, Inc.,2001 3- 24 Irwin/McGraw-Hill Future Values with Compounding Interest Rates

26. © The McGraw-Hill Companies, Inc.,2001 3- 25 Irwin/McGraw-Hill Manhattan Island Sale Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal? To answer, determine $24 is worth in the year 2000, compounded at 8%. FYI - The value of Manhattan Island land is well below this figure.

27. © The McGraw-Hill Companies, Inc.,2001 3- 26 Irwin/McGraw-Hill Present Values Present Value Value today of a future cash flow.

28. © The McGraw-Hill Companies, Inc.,2001 3- 27 Irwin/McGraw-Hill Present Values Present Value Value today of a future cash flow. Discount Factor Present value of a $1 future payment.

29. © The McGraw-Hill Companies, Inc.,2001 3- 28 Irwin/McGraw-Hill Present Values Present Value Value today of a future cash flow. Discount Rate Interest rate used to compute present values of future cash flows. Discount Factor Present value of a $1 future payment.

30. © The McGraw-Hill Companies, Inc.,2001 3- 29 Irwin/McGraw-Hill Present Values

31. © The McGraw-Hill Companies, Inc.,2001 3- 30 Irwin/McGraw-Hill Present Values Example You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

32. © The McGraw-Hill Companies, Inc.,2001 3- 31 Irwin/McGraw-Hill Present Values Discount Factor = DF = PV of $1  Discount Factors can be used to compute the present value of any cash flow.

33. © The McGraw-Hill Companies, Inc.,2001 3- 32 Irwin/McGraw-Hill  The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable. Time Value of Money (applications)

34. © The McGraw-Hill Companies, Inc.,2001 3- 33 Irwin/McGraw-Hill  Value of Free Credit  Implied Interest Rates  Internal Rate of Return  Time necessary to accumulate funds Time Value of Money (applications)

35. © The McGraw-Hill Companies, Inc.,2001 3- 34 Irwin/McGraw-Hill PV of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?

36. © The McGraw-Hill Companies, Inc.,2001 3- 35 Irwin/McGraw-Hill PV of Multiple Cash Flows  PVs can be added together to evaluate multiple cash flows.

37. © The McGraw-Hill Companies, Inc.,2001 3- 36 Irwin/McGraw-Hill Perpetuities & Annuities Perpetuity A stream of level cash payments that never ends. Annuity Equally spaced level stream of cash flows for a limited period of time.

38. © The McGraw-Hill Companies, Inc.,2001 3- 37 Irwin/McGraw-Hill Perpetuities & Annuities PV of Perpetuity Formula C = cash payment r = interest rate

39. © The McGraw-Hill Companies, Inc.,2001 3- 38 Irwin/McGraw-Hill Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

40. © The McGraw-Hill Companies, Inc.,2001 3- 39 Irwin/McGraw-Hill Perpetuities & Annuities Example - Perpetuity In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?

41. © The McGraw-Hill Companies, Inc.,2001 3- 40 Irwin/McGraw-Hill Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

42. © The McGraw-Hill Companies, Inc.,2001 3- 41 Irwin/McGraw-Hill Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

43. © The McGraw-Hill Companies, Inc.,2001 3- 42 Irwin/McGraw-Hill Perpetuities & Annuities PV of Annuity Formula C = cash payment r = interest rate t = Number of years cash payment is received

44. © The McGraw-Hill Companies, Inc.,2001 3- 43 Irwin/McGraw-Hill Perpetuities & Annuities PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.

45. © The McGraw-Hill Companies, Inc.,2001 3- 44 Irwin/McGraw-Hill Perpetuities & Annuities Example - Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

46. © The McGraw-Hill Companies, Inc.,2001 3- 45 Irwin/McGraw-Hill Perpetuities & Annuities Applications  Value of payments  Implied interest rate for an annuity  Calculation of periodic payments  Mortgage payment  Annual income from an investment payout  Future Value of annual payments

47. © The McGraw-Hill Companies, Inc.,2001 3- 46 Irwin/McGraw-Hill Perpetuities & Annuities Example - Future Value of annual payments You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

48. © The McGraw-Hill Companies, Inc.,2001 3- 47 Irwin/McGraw-Hill Inflation Inflation - Rate at which prices as a whole are increasing. Nominal Interest Rate - Rate at which money invested grows. Real Interest Rate - Rate at which the purchasing power of an investment increases.

49. © The McGraw-Hill Companies, Inc.,2001 3- 48 Irwin/McGraw-Hill Inflation approximation formula

50. © The McGraw-Hill Companies, Inc.,2001 3- 49 Irwin/McGraw-Hill Inflation Example If the interest rate on one year govt. bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? Savings Bond

51. © The McGraw-Hill Companies, Inc.,2001 3- 50 Irwin/McGraw-Hill Effective Interest Rates Annual Percentage Rate - Interest rate that is annualized using simple interest. Effective Annual Interest Rate - Interest rate that is annualized using compound interest.

52. © The McGraw-Hill Companies, Inc.,2001 3- 51 Irwin/McGraw-Hill Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

53. © The McGraw-Hill Companies, Inc.,2001 3- 52 Irwin/McGraw-Hill Effective Interest Rates example Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

54. © The McGraw-Hill Companies, Inc.,2001 3- 53 Irwin/McGraw-Hill Web Resources invest-faq.com/articles/analy-fut-prs-val.html www.bankrate.com/brm/default.asp www.financenter.com www.financialplayerscenter.com/Overview.html Click to access web sites Internet connection required