1. 2-1 Copyright © 2006 McGraw Hill Ryerson Limited prepared by: Sujata Madan McGill University Fundamentals of Corporate Finance Third Canadian Edition

2. 2-2 Copyright © 2006 McGraw Hill Ryerson Limited Chapter 4 The Time Value of Money Future Values and Compound Interest Present Values Multiple Cash Flows Perpetuities and Annuities Inflation and Time Value of Money Effective Annual Interest Rates

3. 2-3 Copyright © 2006 McGraw Hill Ryerson Limited Introduction  As a financial manager you will often have to compare cash payments which occur at different dates.  To make optimal decisions, you must understand the relationship between a dollar today [Present value] and a dollar in the future [Future value].

4. 2-4 Copyright © 2006 McGraw Hill Ryerson Limited Future Value  Future value is the amount to which an investment will grow after earning interest.  Interest can be of two types:  Simple interest  Compound interest

5. 2-5 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Simple Interest  Interest is earned only on the original investment.  Example: You invest $100 in an account paying simple interest at the rate of 6% per year. How much will the account be worth in 5 years?

6. 2-6 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Simple Interest  You earn interest only on the original investment  Interest earned per year = $100 x 6% = $6.00  Total interest earned over 5-year period = $6.00 x 5 = $30.00  Balance in account at end of Year 5 = $100 + $30 = $130

7. 2-7 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Simple Interest TodayYear 1Year 3Year 2Year 4Year 5 Starting balance Ending balance Interest earned $100

8. 2-8 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Simple Interest TodayYear 1Year 3Year 2Year 4Year 5 Starting balance Ending balance Interest earned $100 $6 $106

9. 2-9 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Simple Interest TodayYear 1Year 3Year 2Year 4Year 5 Starting balance Ending balance Interest earned $100 $106 $6 $112$118$124$130

10. 2-10 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Compound Interest  Interest is earned on the interest.  Example: You invest $100 in an account paying compound interest at the rate of 6% per year. How much will the account be worth in 5 years?

11. 2-11 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Compound Interest  You earn interest on interest  Interest earned per year = Previous year’s balance x interest rate  Interest earned in Year 1 = $100.00 x 6% = $6.00  Interest earned in Year 2 = $106.00 x 6% = $6.36  Interest earned in Year 3 = $112.36 x 6% = $6.74  Interest earned in Year 4 = $119.10 x 6% = $7.15  Interest earned in Year 5 = $126.25 x 6% = $7.57

12. 2-12 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Compound Interest TodayYear 1Year 3Year 2Year 4Year 5 Starting balance Ending balance Interest earned $100 $106 $6 $6.36 $7.57 $7.15 $6.74 $112.36$119.10$126.25$133.82

13. 2-13 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Compound Interest  Value at the end of Year 1 = $100.00 +[$100 x 6%] = $100 x (1+r)  Value at the end of Year 2 = $106.00 + [$106 x 6%] = $100 x (1+r) 2  …….  Value at the end of Year 5 = $100 x (1+r) 5

14. 2-14 Copyright © 2006 McGraw Hill Ryerson Limited Future Value  In general, value at the end of year t FV = Value today x (1+r) t

15. 2-15 Copyright © 2006 McGraw Hill Ryerson Limited Future Value  Example: You invest $100 in an account paying compound interest at the rate of 6% per year. How much will the account be worth in 5 years? FV = $100 x (1+r) t = $100 x (1+0.06) 5 = $133.82

16. 2-16 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Interest rates

17. 2-17 Copyright © 2006 McGraw Hill Ryerson Limited Future Value Interest rates Note: As r increases, FV increases As t increases, FV increases

18. 2-18 Copyright © 2006 McGraw Hill Ryerson Limited Future Value  Future values are higher because of the interest earned.  This leads to a basic financial principle:  A dollar received today is worth more than a dollar received tomorrow.

19. 2-19 Copyright © 2006 McGraw Hill Ryerson Limited Present Value Let’s turn things around…  How much do you need to invest today into an account paying compound interest at the rate of 6% per year, in order to receive $133.82 at the end of five years?

20. 2-20 Copyright © 2006 McGraw Hill Ryerson Limited Present Value TodayYear 1Year 3Year 2Year 4Year 5 $133.82 ? Future valuePresent value 6%

21. 2-21 Copyright © 2006 McGraw Hill Ryerson Limited Present Value TodayYear 1Year 3Year 2Year 4Year 5 $133.82 ? Present value = $133.82 (1+ 0.06) 5 6% = $100

22. 2-22 Copyright © 2006 McGraw Hill Ryerson Limited Present Value TodayYear 1Year 3Year 2Year 4Year 5 $133.82 ? Present value = $133.82 (1+ 0.06) 5 6% = $100 Simply invert the FV formula to get the PV formula!

23. 2-23 Copyright © 2006 McGraw Hill Ryerson Limited PV = FV after t periods (1+r) t Present Value  In general, present value of a future cash flow:

24. 2-24 Copyright © 2006 McGraw Hill Ryerson Limited Present Value Present value  Example: You have been offered $1 million five years from now. If the interest rates is expected to be 10% per year, how much is this prize worth to you in today’s dollars?

25. 2-25 Copyright © 2006 McGraw Hill Ryerson Limited Present Value TodayYear 1Year 3Year 2Year 4Year 5 $1,000,000 ? Present value = $1,000,000 (1+ 0.10) 5 10% = $620,921

26. 2-26 Copyright © 2006 McGraw Hill Ryerson Limited Present Value vs Future Value PV and FV are related!  The formula for PV is simply the formula for FV inverted! FV at 10% PV at 10% $1,000,000$620,921

27. 2-27 Copyright © 2006 McGraw Hill Ryerson Limited Present Value vs Future Value PV and FV are related! PV = FV x 1/(1 + r) t = $1 million x 1/ (1 + 0.10) 5 = $620,921 FV = PV x (1 + r) t = $620,921 x (1 + 0.10) 5 = $1 million

28. 2-28 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows Future Value  Example: You deposit $1,200 in your bank account today; $1,400 one year later; and $1,000 two years from today. If your bank offers you an 8% interest rate on your account, how much money will you have in the account three years from today?

29. 2-29 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows 0 1 2 $1,000 8% $1,200 8% $1,400 3 8% ? ?? ???

30. 2-30 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows 0 1 2 $1,000 8% $1,200 8% $1,400 3 8% $1,000x(1+0.08) 1 =$1,080.00 $1,400x(1+0.08) 2 =$1,632.96 $1,200x(1+0.08) 3 =$1,511.65 FV = $4,224.61

31. 2-31 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows Present Value  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?

32. 2-32 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows Problem definition  Option 1: $15,500 today  Option 2: $8,000 today; $4,000 at the end of one year; and $4,000 at the end of two years Cash flows can be compared only at the same point in time. Thus, we need to find the PV of Option 2.

33. 2-33 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows Present value of Option 2 0 1 $4,000 8% $8,000 2 8% ? $4,000 ??

34. 2-34 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows Present value of Option 2 0 1 $4,000 8% $8,000 2 8% $4,000/(1+0.08) 1 =$3,703.70 $4,000 $4,000/(1+0.08) 2 =$3,429.36 PV = $15,133.06

35. 2-35 Copyright © 2006 McGraw Hill Ryerson Limited Multiple Cash Flows Compare the two options at the same point in time  PV of Option 1: $15,500.00  PV of Option 2: $15,133.06 Option 1 is better for the buyer. Option 2 is better for the auto dealer.

36. 2-36 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Definitions  Annuity: Equally spaced and a level stream of cash flows [for a finite number of periods]  Perpetuity: Stream of level cash payments that never ends [for an infinite number of periods]

37. 2-37 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Perpetuities  The PV of a perpetuity is calculated by dividing the level cash flow by the interest rate PV of a perpetuity = C r 0 1 3 2……. ∞ C r% CCCCPV

38. 2-38 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Perpetuities  The PV of a perpetuity is calculated by dividing the level cash flow by the interest rate PV of a perpetuity = C r 0 1 3 2……. ∞ C r% CCCC Note: This formula gives you the present value of a perpetuity starting one period from now PV

39. 2-39 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Perpetuities  Example: In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today if the rate of interest is 10%?

40. 2-40 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Perpetuities PV of perpetuity = $100,000 0.10 0 1 … 2……. ∞ 10%…..….10% $100,000 PV $100,000 = $1,000,000

41. 2-41 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Perpetuities  Example contd.: If the first perpetuity payment will not be received until four years from today, how much money needs to be set aside today?

42. 2-42 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Perpetuities PV at end of Year 3 = $100,000 0.10 1 2 4 3 5 ∞ 10% $100,000 PV $100,000 = $1,000,000 …. … PV today = $1,000,000 (1.10) 3 = $751,315 0

43. 2-43 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Annuities  The PV of a t-period annuity is given by: PV of t-period annuity = 0 1 3 2……. t C r% CCCCPV

44. 2-44 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Annuities  The PV of a t-period annuity is given by: PV of t-period annuity = 0 1 3 2……. t C r% CCCCPV Note: This formula gives you the present value of an annuity starting one period from now

45. 2-45 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Annuities  The PV of a t-period annuity is given by: PV of t-period annuity = Note: The term in the square brackets is called the present value annuity factor or PVAF.

46. 2-46 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Annuities  The PV of a t-period annuity is given by: PV of t-period annuity = Note: The term in the square brackets is called the present value annuity factor or PVAF. PVAF can also be computed by using the annuity tables at the end of the book [Appendix A.3].

47. 2-47 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Annuities  Example: You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year, with the first payment one year from now. Given a rate of interest of 10%, what is the price you are paying for the car?

48. 2-48 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities 0 1 2 $4,000 10% $4,000 10% $4,000 3 10% PV of 3-period annuity =

49. 2-49 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Growing Perpetuities PV of a perpetuity = C r - g 0 1 3 2……. ∞ r% C(1+g) 3 C(1+g) 2 C(1+g)CPV

50. 2-50 Copyright © 2006 McGraw Hill Ryerson Limited Perpetuities and Annuities Growing Annuities 0 1 3 2……. t r% C(1+g) 3 C(1+g) 2 C(1+g)CPV

51. 2-51 Copyright © 2006 McGraw Hill Ryerson Limited Inflation Definitions  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.

52. 2-52 Copyright © 2006 McGraw Hill Ryerson Limited Inflation Formulae  The exact formula:  The approximation: 1+ real interest rate = 1+ nominal interest rate 1 + inflation rate Real interest rate ≈ nominal interest rate – inflation rate

53. 2-53 Copyright © 2006 McGraw Hill Ryerson Limited Inflation  Example: If the interest rate on one year government bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? Real interest rate = [(1+ 0.05)/(1+0.022)]-1 = 1.027-1 = 2.7% Real interest rate ≈ 5% - 2.2% = 2.8%

54. 2-54 Copyright © 2006 McGraw Hill Ryerson Limited Effective Annual Interest Rates Definitions  Effective Annual Interest Rate [EAR]: Interest rate that is annualized using compound interest  Annual Percentage Rate [APR]: Interest rate that is annualized using simple interest

55. 2-55 Copyright © 2006 McGraw Hill Ryerson Limited Effective Annual Interest Rates Formulae where m = number of compounding periods in year

56. 2-56 Copyright © 2006 McGraw Hill Ryerson Limited Effective Annual Interest Rates  Example: Given a monthly rate of 1%, what is the Effective Annual Rate [EAR]? What is the Annual Percentage Rate [APR]?

57. 2-57 Copyright © 2006 McGraw Hill Ryerson Limited Effective Annual Interest Rates  Example: Given a monthly rate of 1%, what is the Effective Annual Rate [EAR]? What is the Annual Percentage Rate [APR]? APR = 1% × 12 = 12%

58. 2-58 Copyright © 2006 McGraw Hill Ryerson Limited Summary of Chapter 4  Future value (FV) is the amount to which an investment will grow after earning interest.  The present value (PV) of a future cash payment is the amount you would need to invest today to create that future cash payment.

59. 2-59 Copyright © 2006 McGraw Hill Ryerson Limited Summary of Chapter 4  A level stream of payments which continues forever is called a perpetuity.  One which continues for a limited number of years is called an annuity.  You can use the FV and PV formulas to calculate their value or you can use the shortcut formulae.

60. 2-60 Copyright © 2006 McGraw Hill Ryerson Limited Summary of Chapter 4  Annual percentage rates (APR) do not recognize the effect of compound interest, that is, they annualize assuming simple interest.  Effective Annual Rates (EAR) annualize using compound interest.