Description
We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return
1
The Time Value of
Money
2
What is Time Value?
? We say that money has a time value because that money can
be invested with the expectation of earning a positive rate of
return
? In other words, “a rupee received today is worth more than a
rupee to be received tomorrow”
? That is because today’s rupee can be invested so that we
have more than one rupee tomorrow
3
The Terminology of Time Value
? Present Value - An amount of money today, or the
current value of a future cash flow
? Future Value - An amount of money at some future
time period
? Period - A length of time (often a year, but can be a
month, week, day, hour, etc.)
? Interest Rate - The compensation paid to a lender (or
saver) for the use of funds expressed as a percentage
for a period (normally expressed as an annual rate)
4
Abbreviations
? PV - Present value
? FV - Future value
? Pmt - Per period payment amount
? N - Either the total number of cash flows or
the number of a specific period
? i - The interest rate per period
5
Timelines
0 1 2 3 4 5
PV FV
Today
?A timeline is a graphical device used to clarify the
timing of the cash flows for an investment
?Each tick represents one time period
6
REASONS FOR TIME VALUE OF MONEY
? Risk and Uncertainty
? Inflation
? Consumption:
? Investment opportunities:
7
? EXAMPLE 1: A project needs an initial investment of ` 1,00,000. It is
? expected to give a return of ` 20,000 per annum at the end of each
year, for six years. The project thus involves a cash outflow of `
1,00,000 in the ‘zero year’ and cash inflows of ` 20,000 per year, for six
years.
? In order to decide, whether to accept or reject the project, it is
necessary that the Present Value of cash inflows received annually for
six years is ascertained and compared with the initial investment of `
1,00,000. The firm will accept the project only when the Present Value
of cash inflows at the desired rate of interest exceeds the initial
investment or at least equals the initial investment of ` 1,00,000.
8
Calculating the Future Value
? Suppose that you have an extra saving of Rs.100
today that you wish to invest for one year. If you can
earn 10% per year on your investment, how much
will you have in one year?
0 1 2 3 4 5
100 ?
( )
FV
1
100 1 010 110 = + = .
9
Calculating the Future Value (cont.)
? Suppose that at the end of year 1 you decide to
extend the investment for a second year. How much
will you have accumulated at the end of year 2?
0 1 2 3 4 5
110 ?
( )( )
( )
FV
or
FV
2
2
2
100 1 010 1 010 121
100 1 010 121
= + + =
= + =
. .
.
10
Generalizing the Future Value
? Recognizing the pattern that is developing, we can
generalize the future value calculations as follows:
( )
FV PV i
N
N
= + 1
? If you extended the investment for a third year, you
would have:
( )
FV
3
3
100 1 010 13310 = + = . .
11
Calculating the Present Value
? But we can turn this around to find the amount
that needs to be invested to achieve some desired
future value:
( )
PV
FV
i
N
N
=
+ 1
12
Present Value
? Suppose that your five-year old daughter has just
announced her desire to attend college. After some research,
you determine that you will need about Rs. 100,000 on her
18th birthday to pay for four years of college. If you can earn
8% per year on your investments, how much do you need to
invest today to achieve your goal?
( )
PV = =
100 000
108
769 79
13
,
.
$36, .
13
Annuities
? An annuity is a series of nominally equal payments
equally spaced in time
? Annuities are very common:
• Rent /Car payment
• Pension income
? The timeline shows an example of a 5-year, Rs.100
annuity
0 1 2 3 4 5
100 100 100 100 100
14
The Principle of Value Additives
? How do we find the value (PV or FV) of an annuity?
? First, you must understand the principle of value
additivity:
• The value of any stream of cash flows is equal to the
sum of the values of the components
? In other words, if we can move the cash flows to the same
time period we can simply add them all together to get the
total value
15
Present Value of an Annuity
? We can use the principle of value additivity to find the
present value of an annuity, by simply summing the
present values of each of the components:
( ) ( ) ( ) ( )
PV
Pmt
i
Pmt
i
Pmt
i
Pmt
i
A
t
t
t
N
N
N
=
+
=
+
+
+
+ · · · +
+
=
¿
1 1 1 1
1
1
1
2
2
16
Present Value of an Annuity (cont.)
? Using the example, and assuming a discount rate of 10%
per year, we find that the present value is:
( ) ( ) ( ) ( ) ( )
PV
A
= + + + + =
100
110
100
110
100
110
100
110
100
110
379 08
1 2 3 4 5
. . . . .
.
0 1 2 3 4 5
100 100 100 100 100
62.09
68.30
75.13
82.64
90.91
379.08
17
Present Value of an Annuity (cont.)
? Actually, there is no need to take the present
value of each cash flow separately
? We can use a closed-form of the PV
A
equation
instead:
( )
( )
PV
Pmt
i
Pmt
i
i
A
t
t
t
N
N
=
+
=
÷
+
¸
(
¸
(
(
( =
¿
1
1
1
1
1
18
Present Value of an Annuity (cont.)
? We can use this equation to find the present
value of our example annuity as follows:
( )
PV Pmt
A
=
÷
¸
(
¸
(
(
(
=
1
1
110
010
379 08
5
.
.
.
? This equation works for all regular annuities,
regardless of the number of payments
19
The Future Value of an Annuity
? We can also use the principle of value additivity to find
the future value of an annuity, by simply summing the
future values of each of the components:
( ) ( ) ( )
FV Pmt i Pmt i Pmt i Pmt
A t
N t
t
N
N N
N
= + = + + + + · · · +
÷
=
÷ ÷
¿
1 1 1
1
1
1
2
2
20
The Future Value of an Annuity (cont.)
? Using the example, and assuming a discount rate of 10%
per year, we find that the future value is:
( ) ( ) ( ) ( )
FV
A
= + + + + = 100 110 100 110 100 110 100 110 100 61051
4 3 2 1
. . . . .
100 100 100 100 100
0 1 2 3 4 5
146.41
133.10
121.00
110.00
}
= 610.51
at year 5
21
The Future Value of an Annuity (cont.)
? Just as we did for the PV
A
equation, we could
instead use a closed-form of the FV
A
equation:
( )
( )
FV Pmt i Pmt
i
i
A t
N t
t
N
N
= + =
+ ÷
¸
(
¸
(
(
÷
=
¿
1
1 1
1
? This equation works for all regular annuities,
regardless of the number of payments
22
The Future Value of an Annuity (cont.)
? We can use this equation to find the future
value of the example annuity:
( )
FV
A
=
÷
¸
(
¸
(
(
= 100
110 1
010
61051
5
.
.
.
23
Annuities Due
? Thus far, the annuities that we have looked at begin their
payments at the end of period 1; these are referred to as
regular annuities
? A annuity due is the same as a regular annuity, except that
its cash flows occur at the beginning of the period rather
than at the end
0 1 2 3 4 5
100 100 100 100 100
100 100 100 100 100 5-period Annuity Due
5-period Regular Annuity
24
Present Value of an Annuity Due
? We can find the present value of an annuity due in the
same way as we did for a regular annuity, with one
exception
? Note from the timeline that, if we ignore the first cash flow,
the annuity due looks just like a four-period regular
annuity
? Therefore, we can value an annuity due with:
( )
( )
PV Pmt
i
i
Pmt
AD
N
=
÷
+
¸
(
¸
(
(
(
(
+
÷
1
1
1
1
25
Present Value of an Annuity Due (cont.)
? Therefore, the present value of our example
annuity due is:
( )
( )
PV
AD
=
÷
¸
(
¸
(
(
(
(
+ =
÷
100
1
1
110
010
100 416 98
5 1
.
.
.
? Note that this is higher than the PV of the,
otherwise equivalent, regular annuity
26
Future Value of an Annuity Due
? To calculate the FV of an annuity due, we can
treat it as regular annuity, and then take it one
more period forward:
( )
( )
FV Pmt
i
i
i
AD
N
=
+ ÷
¸
(
¸
(
(
+
1 1
1
0 1 2 3 4 5
Pmt Pmt Pmt Pmt Pmt
27
Future Value of an Annuity Due (cont.)
? The future value of our example annuity is:
( )
( )
FV
AD
=
÷
¸
(
¸
(
(
= 100
110 1
010
110 67156
5
.
.
. .
? Note that this is higher than the future value of
the, otherwise equivalent, regular annuity
28
Deferred Annuities
? A deferred annuity is the same as any other annuity,
except that its payments do not begin until some later
period
? The timeline shows a five-period deferred annuity
0 1 2 3 4 5
100 100 100 100 100
6 7
29
PV of a Deferred Annuity
? We can find the present value of a deferred annuity in the
same way as any other annuity, with an extra step required
? Before we can do this however, there is an important rule to
understand:
When using the PV
A
equation, the resulting PV is always one
period before the first payment occurs
30
PV of a Deferred Annuity (cont.)
? To find the PV of a deferred annuity, we first find
use the PV
A
equation, and then discount that result
back to period 0
? Here we are using a 10% discount rate
0 1 2 3 4 5
0 0 100 100 100 100 100
6 7
PV
2
= 379.08
PV
0
= 313.29
31
PV of a Deferred Annuity (cont.)
( )
PV
2
5
100
1
1
110
010
379 08 =
÷
¸
(
¸
(
(
(
(
=
.
.
.
( )
PV
0
2
379 08
110
31329 = =
.
.
.
Step 1:
Step 2:
32
FV of a Deferred Annuity
? The future value of a deferred annuity is
calculated in exactly the same way as any other
annuity
? There are no extra steps at all
33
Uneven Cash Flows
? Very often an investment offers a stream of cash flows
which are not either a lump sum or an annuity
? We can find the present or future value of such a stream
by using the principle of value additivity
34
Uneven Cash Flows: An Example (1)
? Assume that an investment offers the following cash flows.
If your required return is 7%, what is the maximum price
that you would pay for this investment?
0 1 2 3 4 5
100 200 300
( ) ( ) ( )
PV = + + =
100
107
200
107
300
107
51304
1 2 3
. . .
.
35
Uneven Cash Flows: An Example (2)
? Suppose that you were to deposit the following amounts
in an account paying 5% per year. What would the
balance of the account be at the end of the third year?
0 1 2 3 4 5
300 500 700
( ) ( )
FV= + + = 300 105 500 105 700 155575
2 1
. . , .
doc_806390039.ppt
We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return
1
The Time Value of
Money
2
What is Time Value?
? We say that money has a time value because that money can
be invested with the expectation of earning a positive rate of
return
? In other words, “a rupee received today is worth more than a
rupee to be received tomorrow”
? That is because today’s rupee can be invested so that we
have more than one rupee tomorrow
3
The Terminology of Time Value
? Present Value - An amount of money today, or the
current value of a future cash flow
? Future Value - An amount of money at some future
time period
? Period - A length of time (often a year, but can be a
month, week, day, hour, etc.)
? Interest Rate - The compensation paid to a lender (or
saver) for the use of funds expressed as a percentage
for a period (normally expressed as an annual rate)
4
Abbreviations
? PV - Present value
? FV - Future value
? Pmt - Per period payment amount
? N - Either the total number of cash flows or
the number of a specific period
? i - The interest rate per period
5
Timelines
0 1 2 3 4 5
PV FV
Today
?A timeline is a graphical device used to clarify the
timing of the cash flows for an investment
?Each tick represents one time period
6
REASONS FOR TIME VALUE OF MONEY
? Risk and Uncertainty
? Inflation
? Consumption:
? Investment opportunities:
7
? EXAMPLE 1: A project needs an initial investment of ` 1,00,000. It is
? expected to give a return of ` 20,000 per annum at the end of each
year, for six years. The project thus involves a cash outflow of `
1,00,000 in the ‘zero year’ and cash inflows of ` 20,000 per year, for six
years.
? In order to decide, whether to accept or reject the project, it is
necessary that the Present Value of cash inflows received annually for
six years is ascertained and compared with the initial investment of `
1,00,000. The firm will accept the project only when the Present Value
of cash inflows at the desired rate of interest exceeds the initial
investment or at least equals the initial investment of ` 1,00,000.
8
Calculating the Future Value
? Suppose that you have an extra saving of Rs.100
today that you wish to invest for one year. If you can
earn 10% per year on your investment, how much
will you have in one year?
0 1 2 3 4 5
100 ?
( )
FV
1
100 1 010 110 = + = .
9
Calculating the Future Value (cont.)
? Suppose that at the end of year 1 you decide to
extend the investment for a second year. How much
will you have accumulated at the end of year 2?
0 1 2 3 4 5
110 ?
( )( )
( )
FV
or
FV
2
2
2
100 1 010 1 010 121
100 1 010 121
= + + =
= + =
. .
.
10
Generalizing the Future Value
? Recognizing the pattern that is developing, we can
generalize the future value calculations as follows:
( )
FV PV i
N
N
= + 1
? If you extended the investment for a third year, you
would have:
( )
FV
3
3
100 1 010 13310 = + = . .
11
Calculating the Present Value
? But we can turn this around to find the amount
that needs to be invested to achieve some desired
future value:
( )
PV
FV
i
N
N
=
+ 1
12
Present Value
? Suppose that your five-year old daughter has just
announced her desire to attend college. After some research,
you determine that you will need about Rs. 100,000 on her
18th birthday to pay for four years of college. If you can earn
8% per year on your investments, how much do you need to
invest today to achieve your goal?
( )
PV = =
100 000
108
769 79
13
,
.
$36, .
13
Annuities
? An annuity is a series of nominally equal payments
equally spaced in time
? Annuities are very common:
• Rent /Car payment
• Pension income
? The timeline shows an example of a 5-year, Rs.100
annuity
0 1 2 3 4 5
100 100 100 100 100
14
The Principle of Value Additives
? How do we find the value (PV or FV) of an annuity?
? First, you must understand the principle of value
additivity:
• The value of any stream of cash flows is equal to the
sum of the values of the components
? In other words, if we can move the cash flows to the same
time period we can simply add them all together to get the
total value
15
Present Value of an Annuity
? We can use the principle of value additivity to find the
present value of an annuity, by simply summing the
present values of each of the components:
( ) ( ) ( ) ( )
PV
Pmt
i
Pmt
i
Pmt
i
Pmt
i
A
t
t
t
N
N
N
=
+
=
+
+
+
+ · · · +
+
=
¿
1 1 1 1
1
1
1
2
2
16
Present Value of an Annuity (cont.)
? Using the example, and assuming a discount rate of 10%
per year, we find that the present value is:
( ) ( ) ( ) ( ) ( )
PV
A
= + + + + =
100
110
100
110
100
110
100
110
100
110
379 08
1 2 3 4 5
. . . . .
.
0 1 2 3 4 5
100 100 100 100 100
62.09
68.30
75.13
82.64
90.91
379.08
17
Present Value of an Annuity (cont.)
? Actually, there is no need to take the present
value of each cash flow separately
? We can use a closed-form of the PV
A
equation
instead:
( )
( )
PV
Pmt
i
Pmt
i
i
A
t
t
t
N
N
=
+
=
÷
+
¸
(
¸
(
(
( =
¿
1
1
1
1
1
18
Present Value of an Annuity (cont.)
? We can use this equation to find the present
value of our example annuity as follows:
( )
PV Pmt
A
=
÷
¸
(
¸
(
(
(
=
1
1
110
010
379 08
5
.
.
.
? This equation works for all regular annuities,
regardless of the number of payments
19
The Future Value of an Annuity
? We can also use the principle of value additivity to find
the future value of an annuity, by simply summing the
future values of each of the components:
( ) ( ) ( )
FV Pmt i Pmt i Pmt i Pmt
A t
N t
t
N
N N
N
= + = + + + + · · · +
÷
=
÷ ÷
¿
1 1 1
1
1
1
2
2
20
The Future Value of an Annuity (cont.)
? Using the example, and assuming a discount rate of 10%
per year, we find that the future value is:
( ) ( ) ( ) ( )
FV
A
= + + + + = 100 110 100 110 100 110 100 110 100 61051
4 3 2 1
. . . . .
100 100 100 100 100
0 1 2 3 4 5
146.41
133.10
121.00
110.00
}
= 610.51
at year 5
21
The Future Value of an Annuity (cont.)
? Just as we did for the PV
A
equation, we could
instead use a closed-form of the FV
A
equation:
( )
( )
FV Pmt i Pmt
i
i
A t
N t
t
N
N
= + =
+ ÷
¸
(
¸
(
(
÷
=
¿
1
1 1
1
? This equation works for all regular annuities,
regardless of the number of payments
22
The Future Value of an Annuity (cont.)
? We can use this equation to find the future
value of the example annuity:
( )
FV
A
=
÷
¸
(
¸
(
(
= 100
110 1
010
61051
5
.
.
.
23
Annuities Due
? Thus far, the annuities that we have looked at begin their
payments at the end of period 1; these are referred to as
regular annuities
? A annuity due is the same as a regular annuity, except that
its cash flows occur at the beginning of the period rather
than at the end
0 1 2 3 4 5
100 100 100 100 100
100 100 100 100 100 5-period Annuity Due
5-period Regular Annuity
24
Present Value of an Annuity Due
? We can find the present value of an annuity due in the
same way as we did for a regular annuity, with one
exception
? Note from the timeline that, if we ignore the first cash flow,
the annuity due looks just like a four-period regular
annuity
? Therefore, we can value an annuity due with:
( )
( )
PV Pmt
i
i
Pmt
AD
N
=
÷
+
¸
(
¸
(
(
(
(
+
÷
1
1
1
1
25
Present Value of an Annuity Due (cont.)
? Therefore, the present value of our example
annuity due is:
( )
( )
PV
AD
=
÷
¸
(
¸
(
(
(
(
+ =
÷
100
1
1
110
010
100 416 98
5 1
.
.
.
? Note that this is higher than the PV of the,
otherwise equivalent, regular annuity
26
Future Value of an Annuity Due
? To calculate the FV of an annuity due, we can
treat it as regular annuity, and then take it one
more period forward:
( )
( )
FV Pmt
i
i
i
AD
N
=
+ ÷
¸
(
¸
(
(
+
1 1
1
0 1 2 3 4 5
Pmt Pmt Pmt Pmt Pmt
27
Future Value of an Annuity Due (cont.)
? The future value of our example annuity is:
( )
( )
FV
AD
=
÷
¸
(
¸
(
(
= 100
110 1
010
110 67156
5
.
.
. .
? Note that this is higher than the future value of
the, otherwise equivalent, regular annuity
28
Deferred Annuities
? A deferred annuity is the same as any other annuity,
except that its payments do not begin until some later
period
? The timeline shows a five-period deferred annuity
0 1 2 3 4 5
100 100 100 100 100
6 7
29
PV of a Deferred Annuity
? We can find the present value of a deferred annuity in the
same way as any other annuity, with an extra step required
? Before we can do this however, there is an important rule to
understand:
When using the PV
A
equation, the resulting PV is always one
period before the first payment occurs
30
PV of a Deferred Annuity (cont.)
? To find the PV of a deferred annuity, we first find
use the PV
A
equation, and then discount that result
back to period 0
? Here we are using a 10% discount rate
0 1 2 3 4 5
0 0 100 100 100 100 100
6 7
PV
2
= 379.08
PV
0
= 313.29
31
PV of a Deferred Annuity (cont.)
( )
PV
2
5
100
1
1
110
010
379 08 =
÷
¸
(
¸
(
(
(
(
=
.
.
.
( )
PV
0
2
379 08
110
31329 = =
.
.
.
Step 1:
Step 2:
32
FV of a Deferred Annuity
? The future value of a deferred annuity is
calculated in exactly the same way as any other
annuity
? There are no extra steps at all
33
Uneven Cash Flows
? Very often an investment offers a stream of cash flows
which are not either a lump sum or an annuity
? We can find the present or future value of such a stream
by using the principle of value additivity
34
Uneven Cash Flows: An Example (1)
? Assume that an investment offers the following cash flows.
If your required return is 7%, what is the maximum price
that you would pay for this investment?
0 1 2 3 4 5
100 200 300
( ) ( ) ( )
PV = + + =
100
107
200
107
300
107
51304
1 2 3
. . .
.
35
Uneven Cash Flows: An Example (2)
? Suppose that you were to deposit the following amounts
in an account paying 5% per year. What would the
balance of the account be at the end of the third year?
0 1 2 3 4 5
300 500 700
( ) ( )
FV= + + = 300 105 500 105 700 155575
2 1
. . , .
doc_806390039.ppt