TimeValueOfMoney

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. 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?
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i)n
FV = 100 (1.06)1 = $106
0 1
PV = -100 FV = 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 sums
If 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
0 1
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?
Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i)n
FV = 100 (1.06)5 = $133.82
0 5
PV = -100 FV = 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?
Calculator Solution:
P/Y = 1 I/Y = 6
N = 5 PV = -100
FV = $133.82
0 5
PV = -100 FV = 133.82

14. Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i/m) m x n
FV = 100 (1.015)20 = $134.68
0 20
PV = -100 FV = 134.68
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?

15. Calculator Solution:
P/Y = 4 I/Y = 6
N = 20 PV = -100
FV = $134.68
0 20
PV = -100 FV = 134.68
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?

16. Mathematical Solution:
(Arithmetic Method)
FV = PV (1 + i/m) m x n
FV = 100 (1.005)60 = $134.89
0 60
PV = -100 FV = 134.89
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?

17. Calculator Solution:
P/Y = 12 I/Y = 6
N = 60 PV = -100
FV = $134.89
0 60
PV = -100 FV = 134.89
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?

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

19. Present Value

20. Mathematical Solution:
(Arithmetic Method)
PV = FV / (1 + i)n
PV = 100 / (1.06)1 = $94.34
PV = -94.34 FV = 100
0 1
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%?

21. Calculator Solution:
P/Y = 1 I/Y = 6
N = 1 FV = 100
PV = -94.34
PV = -94.34 FV = 100
0 1
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%?

22. Mathematical Solution:
(Arithmetic Method)
PV = FV / (1 + i)n
PV = 100 / (1.06)5 = $74.73
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%?
0 5
PV = -74.73 FV = 100

23. Calculator Solution:
P/Y = 1 I/Y = 6
N = 5 FV = 100
PV = -74.73
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%?
0 5
PV = -74.73 FV = 100

24. 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
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?

25. Calculator Solution:
P/Y = 1 N = 5
PV = -5,000 FV = 11,933
I/Y = 19%
0 5
PV = -5000 FV = 11,933
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?

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 / (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. Calculator Solution:
 P/Y = 12 FV = 500
 I/Y = 9.6 PV = -100
 N = 202 months
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?
0 ?
PV = -100 FV = 500

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

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

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

31.  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.
Examples of Annuities:

32. 0 1 2 3
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
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08
.
0
1
)
08
.
1
(
000
,
1
1
)
1
(
3
FV
i
i
PMT
FV
n

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

35. 0 1 2 3
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:
10
.
577
,
2
$
08
.
0
)
08
.
1
(
1
1
000
,
1
)
1
(
1
1
3

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


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

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















PV
PV
i
i
PMT
PV
n

37. Calculator Solution:
P/Y = 1 I/Y = 8
PMT = -1,000 N = 3
PV = $2,577.10
0 1 2 3
1000 1000 1000
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%?

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

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

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

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

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

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

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

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

46. 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.
0 1 2 3
1000 1000 1000

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.”
0 1 2 3
1000 1000 1000

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
)
08
.
1
(
08
.
0
1
)
08
.
1
(
000
,
1
)
1
(
1
)
1
(
3





 







 


FV
i
i
i
PMT
FV
n

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%?
0 1 2 3

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
)
08
.
1
(
08
.
0
)
08
.
1
(
1
1
000
,
1
)
1
(
)
1
(
1
1
3












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
















PV
i
i
i
PMT
PV
n

52. Other Cash Flow Patterns
0 1 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. Mathematically,
(PVIFA i, n ) =
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. When n gets very large,
this becomes zero.
So we’re left with PVIFA =
1 -
1
(1 + i)n
i
1
i

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

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 .08
PV = =
= $125,000

59.  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).
Uneven Cash Flows
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000

60.  Sorry! There’s no quickie for this
one. We have to discount each
cash flow back separately.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows

61.  Sorry! There’s no quickie for this
one. We have to discount each
cash flow back separately.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows

62.  Sorry! There’s no quickie for this
one. We have to discount each
cash flow back separately.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows

63.  Sorry! There’s no quickie for this
one. We have to discount each
cash flow back separately.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows

64.  Sorry! There’s no quickie for this
one. We have to discount each
cash flow back separately.
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000
Uneven Cash Flows

65. period CF PV (CF)
0 -10,000 -10,000.00
1 2,000 1,818.18
2 4,000 3,305.79
3 6,000 4,507.89
4 7,000 4,781.09
PV of Cash Flow Stream: $ 4,412.95
0 1 2 3 4
-10,000 2,000 4,000 6,000 7,000

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 = ( 1 + ) m - 1
quoted rate
m

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

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

71. Annual Percentage Yield
(APY)
 Find the APY for the quarterly loan:
APY = ( 1 + ) m - 1
quoted rate
m
APY = ( 1 + ) 4 - 1
APY = .0808, or 8.08%
.0785
4

72. Annual Percentage Yield
(APY)
 Find the APY for the quarterly loan:
 The quarterly loan is more expensive
than the 8% loan with annual
compounding!
APY = ( 1 + ) m - 1
quoted rate
m
APY = ( 1 + ) 4 - 1
APY = .0808, or 8.08%
.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.

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
0 1 2 3 4 5 6 7 8
$0 0 0 0 40 40 40 40 40
 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 2012. Ia ingin memulai tabungan
ini pada awal tahun 2008. Biaya yang
diperlukan untuk keluar negeri adalah Rp.
150.000.000,-.
 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 % ?