Calculating compound interest: It sounds difficult, but it’s an important skill for managing personal finances or making business decisions.

Join us as we develop the formula to calculate interest on principal plus previously-accrued interest, and explain how to do the calculations – with several examples.

## What is Compound Interest?

Financial interest is an increase in the amount of money on loan or on deposit based on an interest rate over a period of time.

A simple definition is that one calculates the new interest amount based on previous interest, not just the original principal. An example will help define several terms.

After one month,*“I _{1}=P_{0}*r*, which you keep on deposit. The new principal (

*P*is “

_{1})*P*.

_{1}=P_{0}+I_{1}=P_{0}+(P_{0}*r)=P_{0}*(1+r)This is “*simple interest*” because we haven’t multiplied any interest amount by the interest rate. That changes in the second month, however.

In the second month, “*I _{2}=P_{1}*r*” and the new principal is

*“P*

_{2}=(P_{1})+(I_{2})=(P_{0}*(1+r))+(P_{1}*r)=(P_{0}*(1+r))+((P_{0}*(1+r)*r))*=(P*“.

_{0}+P_{0}*r)+((P_{0}*r)+(P_{0}*r*r))=P_{0}*((1+r)^{2})That’s pretty messy, but the formula for ‘* t*‘ time periods is “

*P*“.

**=P**_{t}_{0}*((1+r)**)**^{t}

Here is the example in dollars and cents:

- Deposit $1000 at 3% per year.
- On the first anniversary, “
*Interest = $1000*0.03 = $30.*” a - “
*New Principal = $(1000+30) = $1030.*“ - On the second anniversary, “
*Interest = $1030*0.03 = $30.90.*“ - “
*New Principal = $(1030+30.90) = $1060.90.*“

Each year, (assuming no withdrawals) since it’s annual interest, interest is calculated on the current principal, including any accrued interest. (This differs from simple interest, in which accrued interest is not added to the principal when calculating new interest amounts)

- On the third anniversary, use the “
” formula The “**t**time periods*New Principal = $1000*((1.03)*” after rounding up; it all works out the same.^{3}) =”$1092.73

Decoding Science. One article at a time.