Recall
that with simple interest, any accrued interest on a principal amount *P*
is paid once at the end of the compounding period. The **accumulated amount **available at this time would be

**Example.** A bank pays simple interest
at the rate of 8% per year for certain deposits. If a customer deposits $1000
and makes no withdrawals for three years, what is the total amount on deposit
at the end of three years? What is the interest earned in that period of time?

Investigate the annual compounding of $100 over a three-year period with an interest rate of 10% by filling in the following chart.

It
seems that after *t* years, the
accumulated amount is given by the formula

With
compound interest, earned interest is periodically added to the principal to
earn interest itself at the same rate (called the **nominal** rate).

If
the interest is compounded more than once a year (which is typically how things
are done), say *m* times a year, then
the rate per period becomes *r */ *m *and there are a total of *mt* conversion periods in *t* years. The accumulated amount after *t* years is now given by

* *

If $5000 is invested at 7.5% compounded annually, how much is available after 4 years?

If $5000 is invested at 6% compounded semi-annually, how much is available after 12 years?

**How
Often Should We Calculate Interest?**

Consider now an investment of $1000 at 8% for one year, with various compounding frequencies:

Notice that the additional benefit diminishes.

If
a principal *P* is invested at a rate *r* that is compounded **continuously**, then the amount available
after *t* years is given by

$10000 is invested at 10% continuously compounded interest. How much is available after 6 years?

How much should a 35 year old woman invest at 10% continuously compounded interest in order to have $20000 at age 65?

Given
a nominal interest rate of 8% compounded continuously, what is the *effective* interest rate?

How long would it take for an initial investment to double in size? We can’t effectively answer this question just yet.