THE TIME VALUE OF
MONEY
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 todays rupee can be invested so
that we have more than one rupee tomorrow
5 Models of Time Value of
Money
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)
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
Timelines
A timeline is a graphical device used to clarify
the timing of the cash flows for an investment
Each tick represents one time period
PV
0
Today
FV
1
Calculating the Future Value
Suppose that you have an extra 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?
-100
1
FV
1
100 1
2
0.
10
110
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?
-110
?
0
FV2 100 1 0.10 1 0.10 121
or
2
FV2 100 1 0.10
121
Generalizing the Future
Value
Recognizing the pattern that is
developing, we can generalize the
future value calculations as follows:
FVN
PV 1 i
If you extended the investment for a
third year, you would have:
FV3
10 0 1 0.10
1 33.
10
Calculating the Present
Value
So far, we have seen how to calculate
the future value of an investment
But we can turn this around to find the
amount that needs to be invested to
achieve some desired future value:
PV
FVN
1 i
Present Value: An Example
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
1.08
13
$36,769.79
Annuities
An annuity is a series of nominally equal
payments equally spaced in time
Annuities are very common:
Rent
Mortgage payments
Car payment
Pension income
The timeline shows an example of a 5-year,
Rs.100 annuity
100
100
100
100
100
The Principle of Value
Additivity
How do we find the value (PV or FV) of
an annuity?
First, you must understand the principle
of value additivity:
if we can move the cash flows to the same
time period we can simply add them all
together to get the total value
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:
N
PVA
t 1
Pmt t
Pmt 1
Pmt 2
Pmt
N
N
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:
PVA
62.0
68.3
9
75.1
0
82.6
3
90.9
4
1
379.0
8
1 00
1
.
10
10 0
110
.
1 00
1
.
1 0
10 0
1
.
1 0
1 00
1.10
37 9.08
100
100
100
100
100
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 PVA
equation instead:
N
PVA
1 i
t 1
Pmt t
t
Pmt
1 i
Present Value of an Annuity
(cont.)
We can use this equation to find the
present value of our example annuity
as follows:
1
PVA Pmt
.
110
379.08
0.10
This equation works for all regular
annuities, regardless of the number of
payments
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:
FVA
t 1
Pmt
Nt
Pmt
N 1
Pmt
N 2
P mt
The Future Value of an Annuity
(cont.)
FV
A
Using the example, and assuming a discount
rate of 10% per year, we find that the future
value is:
1 00 1
.
1 0
10 0 1
.
10
10 0 1
.
1 0
1 00 1
.
1 0
1 00
61 0.
51
146.41
133.10
121.00
110.00
100
100
100
100
100
=
610.51
at year
5
The Future Value of an Annuity
(cont.)
Just as we did for the PVA equation, we
could instead use a closed-form of the
FVA equation:
N
FVA
Pmt
t 1
Nt
Pmt
This equation works for all regular
annuities, regardless of the number of
payments
The Future Value of an Annuity
(cont.)
We can use this equation to find the
future value of the example annuity:
FVA 100
.
110
0.10
610.51
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
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?
100
0
PV
100
1.07
200
1.07
300
1.07
200
2
513.04
300
3
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?
300
0
FV
300 1
.05
500
1
2
500 1
.05
700
1
,555
.75
700
3
