# Continuous compounding versus discrete compounding

Discrete compounding and continuous compounding are closely related terms. An interest rate is discretely compounded whenever it is calculated and added to the principal at specific intervals (such as annually, monthly or weekly). Continuous compounding uses a natural log-based formula to calculate and add back accrued interest at the smallest possible intervals.

Interest can be compounded discretely at many different time intervals. The number of and distance between compounding periods is explicitly defined with discrete compounding. For example, interest that compounds on the first day of every month is discrete.

There is only one way to perform continuous compounding – continuously. The distance between compounding periods is so small (smaller than even nanoseconds) that it is mathematically equal to zero.

Compounding is still considered to be discrete if it occurs every minute or even every single second. If it isn't continuous, it's discrete. Simple interest is considered discrete as well.

## Calculating Discrete Compounding

If the interest rate is simple – no compounding takes place – then the future value of any investment can be written as: Future Value = principal x ((1 + (interest rate) x (time period)); or, more simply, as FV = P(1+rt)

When interest is compounded discretely, its formula is: FV = P (1+ r/m)^mt, where t is the term of the contract (in years) and m is the number of compounding periods per year.

## Calculating Continuous Compounding

Continuous compounding introduces the concept of the natural logarithm. This is the constant rate of growth for all naturally growing processes. It's a figure that developed out of physics.

The natural log is typically represented by the letter e. To calculate continuous compounding for an interest-generating contract, the formula needs to be written as: FV = P*e^(rt).