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.