The document discusses time value of money concepts and calculations. It explains that time value of money allows comparison of cash flows that occur at different points in time. It then covers key time value of money concepts like future value, present value, compound interest, annuities, and perpetuities. Examples are provided to demonstrate calculations for future value, present value, ordinary annuities, annuities due, and mixed cash flow streams using tables and Excel functions. The document aims to develop an understanding of time value of money principles and computations that are important for accounting, finance, management, and personal financial planning.

1. Principles of
Managerial Finance
Time Value of Money
MBA 656 FINANCIAL MANAGEMENT CYCLE 1
BY: MARY ROSE HABAGAT
GELITA COLON

2. WHY THIS TOPIC MATTERS
TO YOU
IN PROFESSIONAL LIFE:
ACCOUNTING:
You need to understand time-value-of-money
calculations to account for certain transactions
such as loan amortization, lease payments, and
bond interest rates.

3. INFORMATION SYSTEM:
You need to understand time-value-of-money
calculations to design systems that accurately
measure and value the firm’s cash flows.
MANAGEMENT:
You need to understand time-value-of-money
calculations so that you can manage cash
receipts and disbursements in a way that will
enable the firm to receive the greatest value
from its cash flows.

4. MARKETING
You need to understand time value of money
because funding for new programs and products
must be justified financially using time-value-ofmoney techniques.
OPERATIONS
You need to understand time value of money
because the value of investments in new
equipment, in new processes, and in inventory will
be affected by the time value of money.

5. IN YOUR PERSONAL LIFE
Time value techniques are widely used in
personal financial planning. You can use them
to calculate the value of savings at given future
dates and to estimate the amount you need
now to accumulate a given amount at a future
date.
You also can apply them to value lump-sum
amounts or streams of periodic cash flows and
to the interest rate or amount of time needed to
achieve a given financial goal.

6. Learning Objectives
• Discuss the role of time value in finance and the use
of computational aids used to simplify its application.
• Understand the concept of future value, its calculation
for a single amount, and the effects of compounding
interest more frequently than annually.
• Find the future value of an ordinary annuity and an
annuity due and compare these two types of annuities.
• Understand the concept of present value, its
calculation for a single amount, and its relationship to
future value.

7. Learning Objectives
• Calculate the present value of a mixed stream of cash
flows, an annuity, a mixed stream with an embedded
annuity, and a perpetuity.
• Describe the procedures involved in:
– determining deposits to accumulate a future sum,
– loan amortization, and
– finding interest or growth rates

8. The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Time value of money allows comparison of cash flows
from different periods.
Question?
Would it be better for a company to invest
$100,000 in a product that would return a total of
$200,000 in one year, or one that would return
$500,000 after two years?

9. The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Time value of money allows comparison of cash flows
from different periods.
Answer!
It depends on the interest rate!

10. Present Value and Future Value
PRESENT VALUE
FUTURE VALUE
•
Is the cash on hand today
•
•
It is the amount you need
today in to reach a future value
•
•
PRESENT VALUE
TECHNIQUE uses
discounting to find its present
valueof each cash flow at time
zero and then sums these
values to find the investment’s
value today
•
Is cash you will receive at a
given future date
It is the amount you will
receive in the future from your
cash on hand
FUTURE VALUE TECHNIQUE
uses compounding to find
future value of each cash flow
at the end of the investment’s
life and then sums these
values to find the investment’s
future value

11. ILLUSTRATION

12. Simple Interest
With simple interest, you don’t earn interest on
interest.
• Year 1: 5% of $100 =
$5 + $100 = $105
• Year 2: 5% of $100 =
$5 + $105 = $110
• Year 3: 5% of $100 =
$5 + $110 = $115
• Year 4: 5% of $100 =
$5 + $115 = $120
• Year 5: 5% of $100 =
$5 + $120 = $125

13. Compound Interest
With compound interest, a depositor earns interest
on interest!
• Year 1: 5% of $100.00 = $5.00 + $100.00 = $105.00
• Year 2: 5% of $105.00 = $5.25 + $105.00 = $110.25
• Year 3: 5% of $110.25 = $5 .51+ $110.25 = $115.76
• Year 4: 5% of $115.76 = $5.79 + $115.76 = $121.55
• Year 5: 5% of $121.55 = $6.08 + $121.55 = $127.63

14. Computational Aids
• Use the Equations
• Use the Financial Tables
• Use Financial Calculators
• Use Spreadsheets

15. Computational Aids
Future value interest factor or present value interest factor

16. Computational Aids

17. Time Value Terms
• PV0
=
present value or beginning amount
• k
=
interest rate
• FVn
=
future value at end of “n” periods
• n
=
number of compounding periods
• A
=
an annuity (series of equal payments or
receipts)

18. Four Basic Models
• FVn
=
PV0(1+k)n
=
PV(FVIFk,n)
• PV0
=
FVn[1/(1+k)n]
=
FV(PVIFk,n)
A (1+k)n - 1
k
=
A(FVIFAk,n)
= A 1 - [1/(1+k)n] =
A(PVIFAk,n)
• FVAn =
• PVA0
k

19. BASIC PATTERNS OF CASH FLOW
• SINGLE AMOUNT: a lump sum amount
either currently held or expected at some
future date
• ANNUITY: a level periodic stream of cash
flow
• MIXED STREAM: a stream of unequal cash
flows that reflect no particular pattern

20. Future Value Example
Algebraically and Using FVIF Tables
You deposit $2,000 today at 6%
interest. How much will you have in 5
years?
$2,000 x (1.06)5 = $2,000 x FVIF6%,5
$2,000 x 1.3382 = $2,676.40

21. Future Value Example
Using Excel
You deposit $2,000 today at 6%
interest. How much will you have in 5
years?
PV
k
n
FV?
$
2,000
6.00%
5
$2,676
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5, , 2000)

22. Compounding More Frequently
than Annually
• Compounding more frequently than once a year
results in a higher effective interest rate because you
are earning on interest on interest more frequently.
• As a result, the effective interest rate is greater than
the nominal (annual) interest rate.
• Furthermore, the effective rate of interest will increase
the more frequently interest is compounded.

23. Compounding More Frequently
than Annually
• For example, what would be the difference in future
value if I deposit $100 for 5 years and earn 12%
annual interest compounded (a) annually, (b)
semiannually, (c) quarterly, an (d) monthly?
Annually:
100 x (1 + .12)5 =
$176.23
Semiannually:
100 x (1 + .06)10 =
$179.09
Quarterly:
100 x (1 + .03)20 =
$180.61
Monthly:
100 x (1 + .01)60 =
$181.67

24. Compounding More Frequently
than Annually
On Excel
Annually
PV
$
Sem iAnnually Quarterly
100.00
k
12.0%
n
5
FV
$176.23
$
100.00
0.06
10
$179.08
$
100.00
Monthly
$
100.00
0.03
0.01
20
60
$180.61
$181.67

25. Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of 2.7183.
• Continuing with the previous example, find the Future
value of the $100 deposit after 5 years if interest is
compounded continuously.

26. Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
becomes:
FVn (continuous compounding) = PV x (ekxn)
where “e” has a value of 2.7183.
FVn = 100 x (2.7183).12x5 = $182.22

27. Present Value Example
Algebraically and Using PVIF Tables
How much must you deposit today in order to
have $2,000 in 5 years if you can earn 6%
interest on your deposit?
$2,000 x [1/(1.06)5] = $2,000 x PVIF6%,5
$2,000 x 0.74758 = $1,494.52

28. Present Value Example
Using Excel
How much must you deposit today in order to
have $2,000 in 5 years if you can earn 6%
interest on your deposit?
FV
k
n
PV?
$
2,000
6.00%
5
$1,495
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.06, 5, , 2000)

29. Annuities
• Annuities are equally-spaced cash flows of equal size.
• Annuities can be either inflows or outflows.
• An ordinary (deferred) annuity has cash flows that
occur at the end of each period.
• An annuity due has cash flows that occur at the
beginning of each period.
• An annuity due will always be greater than an
otherwise equivalent ordinary annuity because interest
will compound for an additional period.

30. Annuities

31. Future Value of an Ordinary Annuity
Using the FVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the end of each year at 5% interest for
three years.
FVA = 100(FVIFA,5%,3) = $315.25
Year 1 $100 deposited at end of year
=
$100.00
Year 2 $100 x .05 = $5.00 + $100 + $100
=
$205.00
Year 3 $205 x .05 = $10.25 + $205 + $100 =
$315.25

32. Future Value of an Ordinary Annuity
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the end of each year at 5% interest for
three years.
PMT
k
n
FV?
$
100
5.0%
3
$ 315.25
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )

33. Future Value of an Annuity Due
Using the FVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the beginning of each year at 5%
interest for three years.
FVA = 100(FVIFA,5%,3)(1+k) = $330.96
FVA = 100(3.152)(1.05) = $330.96

34. Future Value of an Annuity Due
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much will your deposits grow to if you
deposit $100 at the beginning of each year at 5%
interest for three years.
PMT $ 100.00
k
5.00%
n
3
FV
$315.25
FVA? $ 331.01
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.06, 5,100, )
=315.25*(1.05)

35. Present Value of an Ordinary Annuity
Using PVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes
both principal and interest) at the end of each year for
three years at 10% interest?
PVA = 2,000(PVIFA,10%,3) = $4,973.70

36. Present Value of an Ordinary Annuity
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes
both principal and interest) at the end of each year for
three years at 10% interest?
PMT
I
n
PV?
$
2,000
10.0%
3
$4,973.70
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 3, 2000, )

37. Present Value of an Annuity Due
Using PVIFA Tables
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes
both principal and interest) at the end of each year for
three years at 10% interest?
PVA = 2000(PVIFA,10%,3)(1+k) =
$5,471.40
PVA = 2000(2.487)(1.1) = $5,471.40

38. Present Value of an Annuity Due
Using Excel
• Annuity = Equal Annual Series of Cash Flows
• Example: How much could you borrow if you could
afford annual payments of $2,000 (which includes both
principal and interest) at the end of each year for three
years at 10% interest?
PMT
I
n
PV?
$
2,000
10.0%
3
$5,471.40
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.10, 3, 2000, )

39. Present Value of a Perpetuity
• A perpetuity is a special kind of annuity.
• With a perpetuity, the periodic annuity or cash flow
stream continues forever.
PV = Annuity/k
• For example, how much would I have to deposit today
in order to withdraw $1,000 each year forever if I can
earn 8% on my deposit?
PV = $1,000/.08 = $12,500

40. Future Value of a Mixed Stream
Using Tables
• A mixed stream of cash flows reflects no particular
pattern
• Find the future value of the following mixed stream
assuming a required return of 8%.
Year
Cashflow (1)
No. of years
Year Cash Flow (n) 9%,N(3)
PVIF FVIF
earning int.
1
(2)
400
0.917
Future Value [(1)x(3)]
PV
(4)
$ 366.80
1
2
P11,500
800 = 4
5-1
2
14,0003
5-2
500 = 3
0.8421.360 673.60 P15,640
$
0.7721.260 386.00 17,640
$
3
12,9004
5-3
400 = 2
0.7081.166 283.20 15,041
$
4
16,0005
5-4
300 = 1
0.6501.080 195.00 17,280
$
5
18,000
5-5 = 0
PV
1.000
$1,904.60
Fixed value of mixed stream
18,000
P83,601.40

41. Future Value of a Mixed Stream
Using EXCEL
• Find the present value of the following mixed stream
assuming a required return of 8%.
A
1
B
FUTURE VALUE OF A MIXED STREAM
2
Interest rate, pct/year
8%
3
Year
Year-End Cash
flow
Excel Function
Year Cash Flow
4 1
400 1
800 2
P11,500
P12,900
7 4
500 3
400 4
8 5
300 5
P18,000
5 2
6 3
9NPV
Future
$1,904.76Value
P14,000
P16,000
P83,608.15
Entry in Cell B9
is =FV(B2,A8,0,NPV
(B2,B4:B8)

42. Present Value of a Mixed Stream
Using Tables
• A mixed stream of cash flows reflects no particular
pattern
• Find the present value of the following mixed stream
assuming a required return of 9%.
Year Cash Flow
PVIF9%,N
PV
1
400
0.917
$ 366.80
2
800
0.842
$ 673.60
3
500
0.772
$ 386.00
4
400
0.708
$ 283.20
5
300
0.650
$ 195.00
PV
$1,904.60

43. Present Value of a Mixed Stream
Using EXCEL
• Find the present value of the following mixed stream
assuming a required return of 9%.
A
1
2
3
4
5
6
7
8
9
B
PRESENT VALUE OF A MIXED STREAM OF
CASH FLOWS
Interest rate, pct/year
Year Cash Flow
1
2
3
4
5
9%
Year
Year-End Cash Flow
1
P400
2
P800
3
P500
4
P400
5
P300
400
800
500
400
300
NPVPresent Value
$1,904.76
P1,904.76
Excel Function
Entry in Cell B9 is
=NPV(B2,B4:B8)

44. Compounding Interest More
Frequently Than Annually
• Interest is often compounded more frequently than
once a year. Savings institutions compound interest
semi-annually, quarterly, monthly, weekly, daily, or
even continuously.
SEMIANNUAL COMPOUNDING of interest involves
two compounding periods within the year. Instead of
the stated interest rate being paid once a year, onehalf of the stated interest rate is paid twice a year.
QUARTERLY COMPOUNDING of interest involves four
compounding periods within the year. One-fourth of
the stated interest rate is paid four times a year.

45. Example:
Future Value from Investing P100 at 8% Interest Compounded
Semiannually over 24 Months (2 Years)
Period
Beginning
Principal (1)
Future Value interest
factor (2)
Future value at end
of period [(1)x(2)]
(3)
6 months
P100.00
1.04
P104.00
12 months
104.00
1.04
108.16
18 months
108.16
1.04
112.49
24 months
112.49
1.04
116.99

46. Example:
Future Value from Investing P100 at 8% Interest Compounded
Quarterly over 24 Months (2 Years)
Period
Beginning
Principal (1)
Future Value interest
factor (2)
Future value at end
of period [(1)x(2)]
(3)
3 months
P100.00
1.02
P102.00
6 months
102.00
1.02
104.04
9 months
104.04
1.02
106.12
12 months
106.12
1.02
108.24
15 months
108.24
1.02
110.41
18 months
110.40
1.02
112.62
21 months
112.61
1.02
114.87
24 months
114.86
1.02
117.17

47. Example:
Future Value at the End of Years 1 and 2 from Investing P100 at
8% Interest, Given Various Compounding Periods
Compounding Period
End of Year
Annual
Semiannual
Quarterly
1
P108.00
P108.16
P108.24
2
116.64
116.99
117.17
As shown, the more frequently interest is
compounded, the greater the amount of
money accumulated. This is true for any
interest rate for any period of time.

48. • FVIFi,n
= (1+i/m)mxn
• The basic equation for future value can
no w be rewritten as
FVIFi,n
= (1+i/m)mxn

49. USING COMPUTATIONAL TOOLS FOR
COMPOUNDING MORE FREQUENTLY
THAN ANNUALLY
• Semiannual
Quarterly
Input
100
Function
Input
Function
PV
100
PV
4
N
8
N
4
I
2
I
Solution
is 116.99
CPT
FV
Solution
is 117.17
CPT
FV

50. Spreadsheet Use
A
1
B
FUTURE VALUE OF A SINGLE AMOUNT WITH SEMIANNUAL AND
QUARTERLY COMPOUNDING
2
Present value
3
Interest rate, pct per year compounded semiannually
4
Number of years
5
Future value with semiannual compounding
6
Present value
7
Interest rate, pct per year compounded quarterly
8
Number of years
9
Future value with quarterly compounding
Entry in cell B5 is = FV(B3/2,B4*2,0)
Entry in cell B9 is = FV(B7/4,B8*4,0,-B2,0)
P100
8%
2
P116.99
P100
8%
2
P117.17

51. Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
becomes:
FVn (continuous compounding) = PV x (eixn)
where “e” has a value of 2.7183.
• Continuing with the previous example, To find the value at the
end f 2 years of Fred Moreno’s P100 deposit in an account
paying 8% annual interest compounded continuously

52. Continuous Compounding
• With continuous compounding the number of
compounding periods per year approaches infinity.
• Through the use of calculus, the equation thus
becomes:
FVn (continuous compounding) = PV x (eixn)
where “e” has a value of 2.7183.

53. Continuous Compounding
• CALCULATOR USE
Input
Function
0.16
2nd
1.1735
100
x
=
Solution is 117.35

54. Continuous Compounding
• Spreadsheet Use
A
1
B
FUTURE VALUE OF SINGLE AMOUNT WITH
CONTINOUS COMPOUNDING
2
Present value
P100
3
Annual rate of interest, compounded
continously
8%
4
Number of years
2
5
Future value with continuous compounding
P117.35
Entry in Cell B5 is =B2*EXP(B3*B4)

55. Nominal & Effective Rates
• The nominal interest rate is the stated or contractual
rate of interest charged by a lender or promised by a
borrower.
• The effective interest rate is the rate actually paid or
earned.
• In general, the effective rate > nominal rate whenever
compounding occurs more than once per year
EAR = (1 + i/m) m -1

56. Nominal & Effective Rates
• For example, what is the effective rate of interest on
your credit card if the nominal rate is 18% per year,
compounded monthly?
EAR = (1 + .18/12) 12 -1
EAR = 19.56%

57. Special Applications of Time Value
Future value and present value techniques
have a number of important applications in
finance. We’ll study four of them in this
section:
1.Determining deposits needed to accumulate
a future sum.
2.Loan amortization
3.Finding interest or growth rates, and
4.Finding an unknown number of periods

58. Determining Deposits Needed to
Accumulate a Future Sum
Supposed you want to buy a house 5 years from now,
and you estimate that an initial down payment of
P30,000 will be required at that time. To accumulate
the P30,000, you will wish to make equal annual end-ofyear deposits into an account paying annual interest of
6 percent.
FVAn = PMT X (FVIFAi,n)
PMT = FVAn
FVIFAi,n
FVIFAi,n)
= 1x[ (1+i)n – 1]
i

59. Determining Deposits Needed to
Accumulate a Future Sum
• Calculator Use
Input
3000
Function
FV
5
N
6
I
Solution
is
5,321.89
CPT
PMT

60. Determining Deposits Needed to
Accumulate a Future Sum
Spreadsheet Use
A
1
B
ANNUAL DEPOSITS NEEDED TO ACCUMULATE A FUTURE
SUM
2
Future value
3
Number of years
4
Annual rate of interest
5
Annual deposit
Entry in Cell B5 is =-PMT(B4,B3,0,B2).
Table Use: Use Table A-3
P30,000
5
6%
P5,321.89

61. Loan Amortization
 The term loan amortization refers to the
determination of equal periodic loan
payments.
 Lenders use a loan amortization schedule to
determine these payment amounts and the
allocation of each payment to interest and
principal.
 Amortizing a loan actually involves creating
an annuity out of a present amount.

62. Loan Amortization
You borrow P6000 at 10 percent and agree to
make equal annual end of year payments over
4 years.
PVAn = PMT X (FVIFAi,n)
PMT = PVAn
PVIFAi,n
PVIFAi,n
= 1x[ 1 - 1
(1+i)n
]

63. Loan Amortization
• Calculator Use
Input
Function
6000
PV
4
N
10
I
Solution
is
1,892.82
CPT
PMT

64. Loan Amortization

65. Loan Amortization
A
1
B
ANNUAL PAYMENT TO REPAY A LOAN
2
Loan Principal (present value)
3
Annual rate of interest
4
Number of years
4
5
Annual payment
P1,892.82
Entry cell B5 is = -PMT(B3,B4,B2)
P6,000
10%

66. A
1
2
B
C
Loan Amortization
Data: Loan
Principal
D
E
P6000
3
Annual rate of interest
10%
4
Number of years
4
5
Annual Payments
6
Year
Total
To interest
To Principal
Year-End
Principal
7
0
8
1
1892.82
600.00
1,292.82
4,707.18
9
2
1892.82
470.72
1,422.11
3,285.07
10
3
1892.82
328.51
1,564.32
1,720.75
11
4
1892.82
172.07
1,720.75
0
6,000
Key Cell Entries
Cell B8:=-PMT($D$3,$D$4,$D$2),copy t B9;B11
Cell C8:=-CUMIPMT($D$3,$D$4,$D$2,A8,A8,0), copy to C9:C11
CellD8:=-CUMPRINC($D$3,$D$4,$D$2,A8,A8,0),copy to D9:D11
Cell E8:=E7-D8,copy to E9:E11

67. Loan Amortization
• Use Table A-4

68. Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?
1994 $ 1,000
1995
1,127
1996
1,158
1997
2,345
1998
3,985
1999
4,677
2000
5,525
It is first important to note
that although there are 7
years show, there are only 6
time periods between the
initial deposit and the final
value.

69. Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?
1994 $ 1,000
1995
1,127
1996
1,158
1997
2,345
1998
3,985
1999
4,677
2000
5,525
PV
FV
n
k?
$
$
1,000
5,525
6
33.0%

70. Determining Interest or Growth Rates
• At times, it may be desirable to determine the
compound interest rate or growth rate implied by a
series of cash flows.
• For example, you invested $1,000 in a mutual fund in
1994 which grew as shown in the table below?
1994 $ 1,000
1995
1,127
1996
1,158
1997
2,345
1998
3,985
1999
4,677
2000
5,525
Excel Function
=Rate(periods, pmt, PV, FV)
=Rate(6, ,1000, 5525)

71. Finding an unknown Number of
Periods
• Ann Bates wishes to determine the number of years it
will take for her initial P1000 deposit, earning 8% annual
interest, to grow to equal P2,500. Simply stated, at an
8% annual rate of interest, how many years, n will it take
for Ann’s P1000,PV, to grow to P2,500,FV?
• Table Use:
• We begin by dividing the amount deposited in the
earliest year by the amount received in the latest year.
This will result to present value interest factor
• Use Table A-2

72. Finding an Unknown Number
of Periods
A
B
1
YEARS FOR A PRESENT VALUE TO GROW TO A SPECIFIED
FUTURE VALUE
2
Present value (deposit)
3
Annual Rate of Interest, compounded annually
4
Future value
2,500
5
Number of years
11.91
Entry in Cell B5 is =NPER(B3,0,B2,-B4).
P1000
8%