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The Effect of Compounding
Computing the Effect of Compounding
A General Formula
We have already seen two examples of compounded interest, although in those Jane had to do the compounding herself. As another example, suppose you deposit $1000 at 5% for a period of 2 years and that it is compounded every 6 months. Then the interest paid at the end of each six month period is one-half of 5% of the balance at the beginning of the period. This reflects the fact that 5% is an annual interest rate and six months is one-half of a year. Here is what happens to your balance:
Elapsed Time | Beginning Account Balance | Interest Paid | Ending Account Balance |
---|---|---|---|
6 months | $1000 | 1/2 .05 $1000 = $25 | $1025 |
12 months | $1025 | 1/2 .05 $1025 = $25.62 | $1050.62 |
18 months | $1050.62 | 1/2 .05 $1050.62 = $26.27 | $1076.89 |
24 months | $1076.89 | 1/2 .05 $1076.89 = $26.92 | $1103.81 |
At the end of 2 years you have $1103.81. Had your interest not been compounded, you would have had only $1000 plus 2 5% $1000 which is $1100, so you came out $3.81 ahead.
Financial advisors and books on personal finance often point out the "miracle" of compound interest. This "miracle" is one that takes place over longer periods of time. For example, if you invest your money at a 7% interest rate and simply compound the interest earned once per year, after 10 years instead of having earned interest worth 70% of your original principal, you have earned interest worth 96.71% of your original principal.
Suppose we start with a balance of B dollars and we receive 5% interest compounded every 6 months. This is exactly the situation in the example above, with B = 1000. Then at the end of 6 months the balance (in dollars) will be
5% | ||
B [the original balance] + | [half of the interest rate] B [the original balance]. | |
2 |
Factoring out B, this can be expressed as
The neat thing about this formula is that it represents
the new balance after any 6-month period during which 5%
interest is paid. So to see how much is in the bank after one year,
we can think of the year as being comprised of two 6-month
chunks. If we deposit B dollars at an annual interest
rate of 5% compounded every 6 months, then after 6 months the
balance is
Then the year is divided into n equal time periods during which we earn r/n percent interest. If you start one of these time periods with a balance of B dollars, you will end it with a balance of
dollars, so if you leave your money in the bank for one
year, your balance will grow to
If you deposit B dollars for y years at an
interest rate of r compounded n times per year, then your ending
balance is
times B dollars. |
Example. Suppose you deposit $1000 in a bank which pays 5% interest compounded daily, meaning 365 times per year. How much more do you earn as opposed to simple interest of 5% if you leave your money in the bank for 1 year? For 5 years?
Solution. Referring to the formula
above, the interest rate r is 5% or .05 and the number of equal time
periods, n, is 365. Thus at the end of 1 year your balance is
If you leave the money in the bank for 5 years, then the
formula becomes
You will be best off using your own calculator for this, but we have supplied a clumsy one which you can use as an aid. To use it, type numbers into the first two slots, select an operation (addition (+), multiplication (X), division (/), or raising to a power (^)) and click on the "equals" button. You can cut and paste results into the first two slots as needed.
© 1997,1998 Robby Robson, Oregon State University.
Last Modified: 11/27/2024 23:32:21