Present Value and Future Value - Explanation of the Concept:
Learning Objectives:
- Understand present value concepts and
the use of present value tables.
- Compute the present value of a single
sum and a series of cash flows.
A dollar received now is more valuable than
a dollar received a year from now for the simple reason that if you have a
dollar today, you can put it in the bank an have more than a dollar a year
from now. Since dollars today are worth more than dollars in the future, we
need some means of weighing cash flows that are received at different times
so that they can be compared. Mathematics provides us with the means of
making such comparisons. With a few simple calculations, we can adjust the
value of a dollar received any number of years from now so that it can be
compared with the value of a dollar in hand today. The Mathematics
of Interest:
If a bank pays 5% interest, than a deposit of $100 today will be worth
$105 one year from now. This can be expressed in mathematical terms by means
of the following formula or equation: Formula or Equation:
F1 = P ( 1 +
r ) Where: F1
= the balance at the end of one period, P = the amount invested now, and r =
the rate of interest per period. Example:
If the investment made now is $100
deposited in a bank saving account that is to earn interest at 5%, than P =
$100 and r = 0.05. Under these conditions, F1
= $105, the amount to be received in one year.
The $100 present outlay is called the present
value of the $105 amount to be received in one year. It is also known as the
discounted value of the future $105 receipt. The $100 figure represents the
value in present terms of $105 to be received a year from now when the
interest rate is 5%.
Compound Interest:
When if the $105 is left in the bank for a second year? In that case, by
the end of the second year the original $100 deposit will have grown to
$110.25:
| Original deposit |
$100.00 |
| Interest for the first
year ($100 × 0.05) |
5.00 |
| |
------- |
| Balance at the end of the
first year |
105.00 |
| Interest for the second
year ($105 × 0.05) |
5.25 |
| |
------- |
| Balance at the end of the
second year |
$110.25 |
| |
====== |
Notice that the interest for the second
year is $5.25, as compared to only $5.00 for the first year. The reason for
the greater interest earned during the second year is that during second,
interest is being paid on interest. That is, the $5.00 interest
earned during the first year has been left in the account and has been added
to the original $100 deposit when computing interest for the second year.
This is known as the compound interest. In this case, the compound is
annual. Interest compounded on a semiannual, quarterly, monthly, or even
more frequent basis. The more frequently compounding is done, the more
rapidly the balance will grow. We can
determine the balance in an account after n periods of compounding using the
following formula or equation:
Fn = P (1 = r)n
(1)
Where n = number of periods.
If n = 2 years and the interest rate is 5% per year,
then the balance in two years will be as follows:
F2 = $100 ( 1 + 0.05 )2
F2 = $110.25 Computation of Present Value:
An investment can be viewed in two
ways. It can be viewed either in terms of its future value or in terms of
its present value. We have seen from our computations above that if we know
the present value of a sum (such as $100 deposit), it is a relatively simple
task to compute the sum's future value in n years by using
equation
Fn = P (1 = r)n.
But what if the the tables are reversed and we know the future value of some
amount but we do not know its present value?
For example, assume that you are to receive
$200 two years from now. You know that the future value of this sum is $200,
since this is the amount that you will be receiving after two years. But
what is the sum's present value - what is it worth right now? The present
value of any sum to be received in the future can be computed by turning
equation Fn = P (1 = r)n.
around and solving for P:
P = Fn / ( 1 + r
)n (2)
In our example, F = $200 (the amount to be
received in future), r = 0.05 (the annual rate of interest), and n=2 (the
number of years in the future that the amount is to be received)
P = $200 / (1 + 0.05)n
P = $200 / (1 + 0.05)2
P = $200 / 1.1025
P = $181.40
As shown by the computation above, the present
value of a $200 amount to be received two years from now is $181.40 if the
interest rate is 5%. In effect, $181.40 received right now is equivalent to
$200 received two years from now if the rate of return is 5%. The $181.40
and the $200 are just two ways of looking at the same thing.
The process of finding the present value of a
future cash flow, which we have just completed, is called discounting. We
have discounted the $200 to its present value of $181.40 The 5% interest
figure that we have used to find this present value is called the discount
rate. Discounting future sums to their present value is a common practice in
business, particularly in capital budgeting decisions.
If you have a power key (yx)
on your calculator, the above calculations are fairly easy. However, some of
the present value formulas will be using are more complex and difficult to
use. Fortunately, tables are available in which many of the calculations
have already been done for you. For example, Table 3 at
Future Value and Present
Value Tables page shows the discounted present value of $1 to be
received two periods from now at 5% is 0.907. Since in our example we want
to know the present value of $200 rather than just $1, we need multiply the
factor in the table by $200:
$200 ×
0.907 = $181.40 This answer is the
same as we obtained earlier using the formula in equation (2).
Present Value of a Series of Cash Flow:
Although some investments involve a single
sum to be received (or paid) at a single point in the future, other
investments involve a series of cash flows. A series (or stream) of
identical cash flows is known as an annuity. To provide an example, assume
that a firm has just purchased some government bonds in order to temporarily
invest funds that are being held for future plant expansion. The bonds will
yield interest of $15,000 each year and will be held for five years. What is
the present value of the stream in interest receipts from the bonds? As
shown from the following calculations the present value of this stream is
$54,075 if we assume a discount rate of 12% compounded annually.
|
Year |
Factor at 12%
(Future Value and Present
Value Tables-Table 3) |
Interest Received |
Present Value |
|
1 |
0.893 |
$15,000 |
$13,395 |
|
2 |
0.797 |
15,000 |
11,955 |
|
3 |
0.712 |
15,000 |
10,680 |
|
4 |
0.636 |
15,000 |
9,540 |
|
5 |
0.567 |
15,000 |
8,505 |
| |
|
|
--------- |
| |
|
|
$54,0785
======= |
The discount factors used in this
calculation have been taken from
Future Value and Present Value Table - Table 3.
Two points are important in connection with
this computation. . First, notice that the present value of the $15,000
received a year from now is $13,395, as compared to only $8,505 for the
$15,000 interest payment to be received five years from now. This point
simply underscores the fact that money has a time value.
The second point is that the computations
involved above involve unnecessary work. The same present value of $54,075
could have been obtained more easily by referring to Table 4 at
Future Value and Present Value Table. Table 4 contains the present value
of $1 to be received each year over a series of years at various interest
rates. This table have been derived by simply adding together the factor
from Table 3 as follows:
The sum of the five factors above is 3.065.
Notice from the Table 4 at
Future Value and Present Value Tables Page that the factor of $1 to be
received each year for five years at 12% is also 3.605. If we use this
factor and multiply it by the $15,000 annual cash inflow, then we get the
same $54075 present value that we obtained earlier.
$15,000 × 3.605 = $54,075
Therefore, when computing the present value of
a series (or stream) of equal cash flows that begins at the end of period 1,
Table 4 should be used. To summarize,
the the present value tables, at
Future Value and Present Value Tables Page, should be used as follows:
Table 3: This table should be used to
find the present value of a single cash flow (such as a single payment or
receipt) occurring in future.
Table 4: This table should be used to find the present value of a series
(or stream) of identical cash flows beginning at the end of the current
period and continuing into the future. |
