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
Bachelor of Economics (Majoring in Finance), Petra Christian University
Postgraduate Diploma in Finance, The University of Melbourne
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. We know that receiving $1 today is worth more than $1 in the future
We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs.
The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.
Today
Future

7. 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).
Today ?
Future ?
Today
Future

8. Future Value

9. Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?
PV = FV = 106
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
I am providing information on the Texas Instruments BA-II Plus – other calculators are similar. If you recommend or require a specific calculator other than this one, you may want to make the appropriate changes.
Note: the more information students have to remember to enter the more likely they are to make a mistake. For this reason, I normally tell my students to set P/Y = 1 and leave it that way. Then I teach them to work on a period basis, which is consistent with using the formulas. If you want them to use the P/Y function, remind them that they will need to set it every time they work a new problem and that CLR TVM does not affect P/Y.
If students are having difficulty getting the correct answer, make sure they have done the following:
Set decimal places to floating point (2nd Format, Dec = 9 enter)
Double check and make sure P/Y = 1
Make sure to clear the TVM registers after finishing a problem (or before starting a problem) It is important to point out that CLR TVM clears the FV, PV, N, I/Y and PMT registers. C/CE and CLR Work DO NOT affect the TVM keys
The remaining slides will work the problems using the notation provided above for calculator keys. The formulas are presented in the notes section.

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

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

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

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

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

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

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

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

19. Present Value

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

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

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

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

24. Present Value - single sums If 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
= ((1/ (1+i)5)
= (1+i)5
(2.3866)1/5 = (1+i)
i = .19

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

26. Present Value - single sums Suppose 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 / ( )N
5 = (1.008)N
ln 5 = ln (1.008)N
ln 5 = N ln (1.008)
= N
N = 202 months

27. Present Value - single sums Suppose 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? ?
PV = FV =
Calculator Solution:
P/Y = 12 FV = 500
I/Y = 9.6 PV = -100
N = months

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

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

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

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. Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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

34. Future Value - annuity If 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, N = 3
FV = $3,246.40

35. Present Value - annuity What 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 - annuity What 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. Present Value - annuity What 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, N = 3
PV = $2,577.10

38. Ordinary Annuity vs. Annuity Due
$ $ $1000

39. Begin Mode vs. End Mode

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

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

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

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

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

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

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

47. What about this annuity?
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. 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. 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%?

51. Present Value - annuity due
Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:
PV = PMT (PVIFA i, n ) (1 + i)
PV = 1,000 (PVIFA .08, 3 ) (1.08) use PVIFA table or
= $2,783.26

52. Other Cash Flow Patterns1 2 3

53. Perpetuities
Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.
You can think of a perpetuity as an annuity that goes on forever.

54. Present Value of a Perpetuity
When we find the PV of an annuity, we think of the following relationship:
PV = PMT (PVIFA i, n )

55. 1 - i Mathematically, 1 (PVIFA i, n ) = (1 + i)
We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large?
1 - 1
(1 + i) n i

56. 1 - 1 i i When n gets very large, this becomes zero. 1 (1 + i)
So we’re left with PVIFA =
1 - 1
(1 + i) n i 1 i

57. Present Value of a Perpetuity
So, the PV of a perpetuity is very simple to find:
PMT i
PV =

58. What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?
PMT $10,000 i
PV = =
= $125,000

59. Uneven Cash Flows -10,000 2,000 4,000 6,000 7,000 Is this an annuity?
-10, , , , ,000
Is this an annuity?
How do we find the PV of a cash flow stream when all of the cash flows are different? (Use a 10% discount rate).

60. Uneven Cash Flows
-10, , , , ,000
Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

61. Uneven Cash Flows
-10, , , , ,000
Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

62. Uneven Cash Flows
-10, , , , ,000
Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

63. Uneven Cash Flows
-10, , , , ,000
Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

64. Uneven Cash Flows
-10, , , , ,000
Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.

65.
-10, , , , ,000
period CF PV (CF)
0 -10, ,000.00
, ,818.18
, ,305.79
, ,507.89
, ,781.09
PV of Cash Flow Stream: $ 4,412.95

66. Annual Percentage Yield (APY)
Which is the better loan:
8% compounded annually, or
7.85% compounded quarterly?
We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year!
We need to calculate the APY.

67. Annual Percentage Yield (APY)

68. Annual Percentage Yield (APY)
APY = ( ) m - 1
quoted rate m

69. Annual Percentage Yield (APY)
APY = ( ) m - 1
quoted rate m
Find the APY for the quarterly loan:

70. Annual Percentage Yield (APY)
APY = ( ) m - 1
quoted rate m
Find the APY for the quarterly loan:
APY = ( )
.0785 4

71. Annual Percentage Yield (APY)
APY = ( ) m - 1
quoted rate m
Find the APY for the quarterly loan:
APY = ( )
APY = .0808, or %
.0785 4

72. Annual Percentage Yield (APY)
APY = ( ) m - 1
quoted rate m
Find the APY for the quarterly loan:
The quarterly loan is more expensive than the 8% loan with annual compounding!
APY = ( )
APY = .0808, or %
.0785 4

73. Spreadsheet Example Use the following formulas for TVM calculations
FV(rate,nper,pmt,pv)
PV(rate,nper,pmt,fv)
RATE(nper,pmt,pv,fv)
NPER(rate,pmt,pv,fv)
The formula icon is very useful when you can’t remember the exact formula
Click on the Excel icon to open a spreadsheet containing four different examples.
Click on the tabs at the bottom of the worksheet to move between examples.

74. Practice Problems
 2002, Prentice Hall, Inc.

75. Example
Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows?

76. Example
Cash flows from an investment are expected to be $40,000 per year at the end of years 4, 5, 6, 7, and 8. If you require a 20% rate of return, what is the PV of these cash flows? $

77. Retirement Example
If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30?

78. House Payment Example
If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

79. Example
Baim ingin menabung untuk biaya wisata keluar negeri. Bajuri ingin keluar negeri pada akhir tahun Ia ingin memulai tabungan ini pada awal tahun Biaya yang diperlukan untuk keluar negeri adalah Rp ,-.
Berapakah yang harus ditabung oleh Baim bila ia ingin menabung sekali saja pada awal tahun 2008 bila suku bunga adalah 15 % ?
Berapakah yang harus ditabung Baim bila ia ingin menabung setiap awal tahun mulai tahun 2008 hingga tahun 2012, asumsi suku bunga 15 % ?