Comprehensive Guide on Compound Interest
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Motivating example
Suppose we initially have
Today | 1st year | 2nd year | 3rd year |
---|---|---|---|
In general, we can see that the new balance for any year is computed by:
Where:
is the bank balance of the -th year. is the bank balance of the previous ( -th) year. is the interest rate in decimals.
But the bank balance of the previous year
Again
We can keep repeating this until we get to the present day, that is, when we have our initial bank balance
This equation gives us the bank balance after
Once again, let's assume that we initially have
Today | 0.5 year | 1 year | 1.5 year |
---|---|---|---|
We can see that the formula to compute the bank account after
Here,
If we keep repeating the same process until we reach the present day, we should end up with:
Notice the following:
the exponent is
instead of because the frequency in which we receive the interest is doubled.we are dividing
by because the interest rate is cut in half.
If the interest is compounded quarterly, then we can guess that the formula is:
This leads us to the general formula for compound interest.
Formula to compute compound interest
The compound interest is calculated by:
Where:
is the final amount in the bank account after years. is the principal, that is, the starting amount in the bank account. is the interest rate (e.g. ). is the number of compounds per year. is the number of years elapsed.
Computing compound interest (1)
Suppose we have
Solution. We are given the following information:
Let's directly apply our formula:
Therefore, we should expect to have roughly
Computing compound interest (2)
Suppose we have
Solution. We are given the following information:
We apply our formula for compound interest:
Therefore, after