# Compound interest formula

Before learning about compound interest and the compound interest formula, make sure you've read the lesson on simple interest.

## What is compound interest?

It's interest calculated both on the principal (the amount of money originally borrowed or invested) and interest accrued during previous periods of the loan or investment.

## Some examples of how to calculate compound interest

In the lesson on simple interest, we met William who borrowed \$1000 from his bank over 2 years at an annual simple interest rate of 5%. This equated to simple interest of \$100, i.e. a total amount owing of 1000 + 100 = \$1100.

But what if William had borrowed \$1000 over 2 years at an annual compound interest rate of 5%?

During the first year, he would have accrued 1000 x 5% = \$50 interest. At the end of the 1st year, the total amount owed would have therefore been 1000 + 50 = \$1050.

In the second year, William would have accrued 1050 x 5% = \$52.50 interest. At the end of the 2nd year, the total amount owed would have been 1050 + 52.50 = \$1102.50.

You can see that borrowing \$1000 over 2 years at an interest rate of 5% incurs an additional \$1102.50 - \$1100 = \$2.50 interest if the loan is based on compound rather than simple interest.

This assumes no repayments in the interim. However, in reality the loan would probably be repaid gradually with the effect of reducing the overall amount of simple or compound interest payable.

The above is an instance of how compound interest can work against you - it puts you in further debt compared to simple interest based on the same percentage rate.

## But compound interest can also work for you!

Take the example of Emma who we also met in the simple interest lesson. She invested \$200 in a savings account with an annual simple interest rate of 3%. After 5 years, her total savings were \$230 (original \$200 plus \$30 simple interest).

If Emma had invested her \$200 in a savings account with an annual compound interest rate of 3%, her money would have grown over 5 years as follows

1. 200 + (200 x 3%) = 200 + 6 = \$206
2. 206 + (206 x 3%) = 206 + 6.18 = \$212.18
3. 212.18 + (212.18 x 3%) = 212.18 + 6.3654 = \$218.5454
4. 218.5454 + (218.5454 x 3%) = 218.5454 + 6.556362 = \$225.101762
5. 225.101762 + (225.101762 x 3%) = 225.101762 + 6.75305286 = \$231.8548149

Emma would have ended up with \$231.85 (rounded down to the nearest cent), an additional \$1.85 in interest.

The additional interest earned and incurred by Emma and William may seem like modest sums, but given time and/or a larger principal, savings or debt quickly compound or snowball into larger amounts of money!

## Isn't there an easier way to calculate compound interest?

Yes - use the compound interest formula where

• p = principal
• r = interest rate per period (expressed as a percentage %)
• n = number of periods

As with simple interest, a typical interest rate is per year and money is borrowed or invested (and interest accrues) over a number of years.

Going back to William, p = \$1000, r = 5%, n = 2 years, so

compound interest   =   1000 (1 + 5%)²  -  1000   =   \$102.50

On your calculator enter 1000, multiply sign (x), left bracket sign, 1, add sign (+), 5, percentage sign (%), right bracket sign, x² sign, equals sign (=), subtract sign (-), 1000, equals sign (=). You may need to access % and x² via a shift or 2nd function button.

In Emma's case, p = \$200, r = 3%, n = 5 years, therefore

compound interest   =   200 (1 + 3%)⁵  -  200   =   \$31.85

rounded down to the nearest cent.

Your calculator key strokes are 200, multiply sign (x), left bracket sign, 1, add sign (+), 3, percentage sign (%), right bracket sign, xy sign (i.e. x to the power of y), 5, equals sign (=), subtract sign (-), 200, equals sign (=)

## Practice using the compound interest formula

Now make up some of your own principal loan/investment and interest rate figures and plug them into the compound interest formula.

Apply the same figures to the simple interest formula and compare the outcomes. You may be surprised what a difference compounding can make!

For instance, say Emma leaves her initial \$200 investment untouched for 25 years - given the same 3% interest rate, she'll earn \$150 in simple interest but \$218.75 if her savings are based on compound interest. Compounding will more than double her original investment!

› Compound interest formula