# Compound interest formula

A simpler version of the compound interest formula is A  = P( 1 + r)t  where A is the final balance, P is the principal, r is the annual interest rate compounded once per year, and t is the time in years. The principal is the amount of money you deposit that you expect will grow over time.

## An example showing how to use the simpler version of the compound interest formula

Example #1

A businessperson invests 20000 dollars in a local bank paying 6% interest every year. How much money does the businessperson have in his account after 8 years?

Solution:

In this scenario, the interest rate is compounded or "calculated and added to the account" only once per year. Therefore, you can just use the formula A  = P( 1 + r)t to find the accumulated amount after 8 years.

A  = P( 1 + r)t

A  = 20000( 1 + 6%)8

A  = 20000( 1 + 0.06)8

A  = 20000( 1.06)8

A  = 2000(1.5938480)

A = 31,876.96

Many banks though have plans in which interest is paid more than once a year. The number of interest periods is the number of times the interest is computed and paid per year.

If the interest is computed and added to the account twice a year, this means that the number of interest periods is 2.

If the interest is computed and added to the account quarterly or four times a year, this means that the number of interest periods is 4.

Suppose the interest rate is 6% per year and the number of interest periods is 4. Then, each time the interest is compounded, the bank will use 6% / 4 or 1.5%.

In general, if r is the yearly interest and n is the number of interest periods in a year, each time the interest is compounded, the bank will use r / n.

The number of payment periods will also change. In the simpler version of the formula shown above, the number of payment periods is t. And t is the number of years.

Suppose each year though the interest is compounded 4 times. After 8 years, the number of payment periods is 4 × 8 or 32.

The number of payment periods is the total number of times interest is added to the account. In this case, it was done 32 times.

In general, if n is the number of interest periods in a year and t is the number of years, then the number of payment periods is n × t.

Therefore, a more complete version of the compound interest formula is:

A  = P( 1 + r / n)nt

## How to use the more complete version of the compound interest formula

Example #2

Let us modify example #1 a little bit!

A businessperson invests 20000 dollars in a local bank paying 6% interest every year. The bank computes interest 4 times per year. How much money does the businessperson have in his account after 8 years?

Solution:

Now the interest rate is compounded or "calculated and added to the account" four times per year.

Therefore, you must use the formula A  = P( 1 + r / n)nt to find the accumulated amount after 8 years.

A  = 20000( 1 + r / n)nt

A  = 20000( 1 + 6% / 4)

A  = 20000( 1 + 1.5%)32

A  = 20000( 1 + 0.015)32

A  = 20000( 1.015)32

A  = 20000(1.61)

A  = 32200

Notice that when the interest is paid 4 times a year, you end up with a little bit more money!

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