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Learning objectives of this
article:
-
Define and explain the future value.
-
How is it calculated?
Contents:
Future value concept into two types.
These are: (1) future value of a
single sum and (2) future value of an annuity. In
this article future value of a single sum is
explained. To understand the concept of the future
value of an annuity read future
value of an annuity article.
To understand the concept of future value we need
to understand compound interest first. Under
the procedure of compounding, the interest is
reinvested. The interest earned each period is added
to the principal for the purpose of compounding
interest for the next period. The amount of interest
computed using this procedure is called the
compound interest. The principle plus any
interest earned during a period is called
Compound amount or future value.
Suppose that we have deposited
$8,000 in a credit union which pays interest of 8
percent per year compounded quarterly. We want to
determine the amount of money we will have on
deposit at the end of 1 year if all interest is left
in the savings account. At the end of the first
quarter, interest is computed as follows:
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I1 =
($8,000)(0.08)(0.25*)
= $160
*1/4 - First
quarter
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With the interest left in the account, the
principal on which interest in the second quarter is
the original principal plus the $160 in interest
earned during the first quarter or $8,000 + $160 =
$8,160. Interest for the second quarter is computed
as follows:
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I2 =
($8,160)(0.08)(0.25*)
= $163.20
*1/4 -
Second quarter
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The calculation for the four quarters is as follows:
|
Quarter |
(P) Principle |
(I) Interest |
(S = P + I) Compound Amount |
| 1 |
$8,000.00 |
$160.00 |
$8,000.00 |
+ |
$160.00 |
= |
$8,160.00 |
| 2 |
$8,160.00 |
$163.00 |
$8,160.00 |
+ |
$163.00 |
= |
$8,323.20 |
| 3 |
$8,323.20 |
$166.46 |
$8,323.20 |
+ |
$166.46 |
= |
$8,489.66 |
| 4 |
$8,489.66 |
$169.79 |
$8,489.66 |
+ |
$169.79 |
= |
$8,659.46 |
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In this example, compound interest for one year
is $659.46 ($160.00 + 163.20 + 166.46 + 169.79). The
total amount at the end of the year (principal plus
interest earned) is $8,659.46. This is compound
amount or future value of the original amount
(principal) of $8,000.
We have already discussed that the
interest earned plus principal is equal to compound
amount or future value. The above method of
computing compound amount is time consuming. To save
time and make the procedure simple, we can use the
following formula:
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S = P(1 + i)n
Where;
- P = Principal, dollars
- i = Interest rate per compounding
period
- n = Number of compounding periods
(number of periods in which the
principal has earned interest)
- S = Compound amount
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Example 1:
Suppose that $1,000 is invested in savings bank
which earns interest at a rate of 8 percent per year
compounded annually. If all interest is left in the
account, what will the account balance be after 10
years.
Solution:
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S = P(1 + i)n
S = $1,000(1 + 0.08)10
S = ($1,000)(2.15892*)
= $2,158.92
The $1,000 investment will grow to
$2,158.92, meaning that interest of
($2,158.92 - $1,000) $1,158.92 will be
earned. In other words the $1,000 will have
a value equal to $2,158.92 after 10 years.
*Future
value of $1 table - (1 + i)n
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Example 2:
A long term investment has been made by a small
company. The interest rate is 12% per year, and
interest is compounded semiannually. If all interest is
reinvested at same rate of interest, what will the
value of the investment be after 8 years?
Solution:
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S = P(1 + i)n
S = $250,000(1 + 0.06)16
S = $250,000(2.54035*)
= $635,087.5
*Future
value of $1 table - (1 + i)n
- Compounding occurs twice a year
(semiannually). The interest rate for
six months is the annual interest rate divided by the
number of compounding periods per year
i.e:
0.12/2
= 0.06
- The number of compounding periods
over the 8-year period is 8
× 2
= 16
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