Compound Interest Formula -its origins
You are now going to meet the compound interest formula. When you meet a
formula for the first time, you should wonder what it does or means. You
should wonder where it came from or how it was obtained.
What is interest?
When you place (or invest) money in a bank account, a bank pays you money
for keeping your money with it, a form of rent for its use of it. The
bank is using your money to make loans or investments. The money you are
paid is called interest. The amount of interest paid and how often
depends on the type of account.
What is Compound Interest
In a compound interest account, a bank adds the interest to your account
at the end of a period. This period may be a day, a month, a quarter
year, a half-year or a full-year. In each period, all the money in your
account is now earning interest. So you now receive interest or rent not
only for your original deposit, but also for interest previously added to
the account on the completion of each period. Here interest paid at the
end of one period will earn interest in future periods. Your money is
said to be earning compound interest, or more briefly, compounding. The
following table (not in the paperback version) is included to review or
introduce how and why the compound interest formulas works.
The Formula
When you place an initial amount P into an account, it is called
the principal. In a compound interest account the following happens. The
money in your account grows to an amount A after n periods.
(The number n here identifies the number of periods your money
stays in the account without any withdrawals, or deposits, except for
interest payments at the end of each period.) The amount A is
given by the compound interest formula
In this formula, the interest rate per period is given by the quantity
i. The formula should only be used when interest is compounded.
Again, compounded means the interest is reinvested at the end of each
period with no other deposits or withdrawals, Each interest payment
deposited in your account then earns interest (rent from the bank) in the
following periods.
Numerical Motivation for the Formula
Here is a numerical examples with i = 5% and P = 1000 to
show how or why the formula works. Observe how the amount at the
end of a period is the same as the amount at the start of the next
period.
Period
n
|
Amount at
Start of Period
|
Amount of Interest
|
Amount at end of Period
|
103(1.05)n
|
|
1
|
1000.00
|
50.00
|
1050.00
|
1050.00
|
|
2
|
1050.00
|
52.50
|
1102.50
|
|
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3
|
1102.50
|
|
|
|
|
4
|
|
|
|
|
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5
|
|
|
|
|
|
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Observe how the amout at the end of a period is
equals 100% of the initial amount plus 5% of the initial
amount. So the amount at the end each period is 105% of 1.05
times the initial amount.
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Fill in this table with the aid of a calculator to
the nearest penny (two decimal places). Observe the formula use
shortens the calculation. Note how the amount at the end of one
period becomes the amount at the start of the next. If you
do not like to work with interest calculations, turn this whole
chapter into a compound population growth model using the values
of A = P(1+i)n to
nearest whole number as an approximation to the whole number of
individuals present in the population.
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The compound interest formula gives an example of a calculation described
in algebraic shorthand notation. To use the compound interest formula
someone has to explain or show to you the role of each piece of the
shorthand. That is done next.
The final compounded amount A on the left-hand side of the
compound interest formula can be computed when three numbers are given,
namely
- the initial amount, also called the principal P,
- the interest rate i, and
- the number n of compounding periods, possibly months, in which
interest is compounded.
When these numbers or quantities are not given, we can only talk or
write about compound interest calculations and not do them. Examples in
which calculations are done and numbers appear are given below.
- Other people thinking perhaps of the word rate rather than the word
interest in the phrase interest rate, use the letter r
instead of i. The shorthand selected does not matter. Like a play,
only the plot is important. The actors or letters can be changed.
We could try to describe the compound interest calculation in words
alone. This description might be a good essay assignment in a language
course alongside the essay of describing in words alone how to tie a
shoelace. The task is formidable. The task should persuade you that the
algebraic shorthand notation has a few space-saving advantages, even if
it may be difficult to read aloud in an understandable way. Formulas like
pictures need to be seen to be fully appreciated. Often, mathematics is
better written and not spoken.
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
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Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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Appetite.
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