You can skip this section if you aren’t interested in all the details of the calculations. I recommend that you at least glance through this section to try and get an idea of what’s going on in various calculations.
At the outset of the loan, the interest is just the principal times the interest rate per compounding interval, therefore we may write
Interest = P—,
where P is the principal, R is the interest rate per year, and n is the number of compounding intervals per year.
To get the new balance, you add this interest to the principal:
Balance = P + P—.
You can see that the principal P appears twice in this equation. The rules of algebra let us write this same formula as
Balance = P | 1 + R
Now, suppose you want to get the balance after the second compounding period. You just repeat what you’ve done, but instead of using the principal, you use the last balance. The new balance is then
Balance = P ^1 + R j^1 + R j.
After three compounding periods, the balance would be
Balance = P
and so on.
Using exponential notation,
where n is the number of compounding intervals.
This formula is useful for calculating the balance after any number of compounding intervals on a pocket calculator because all but the simplest of pocket calculators will have the ability to raise an expression to the power n, as shown.