# Continuously compounding interest

Author:
Mark

## Compound Interest

The formula for computing compound interest is , where is the final amount, is the initial principal balance, is the rate for a unit of time, is the number of times compounded in one unit of time, and is the time elapsed.

What is the amount in a bank account that initially has \$1000 and has a yearly interest rate of 2% compounding quarterly after 2 years?

Check all that apply
But what happens to the output value as the interest compounds more and more frequently? We let ,, and for simplicity.

What appears to happen to the compound interest?

## Deriving a continuous compound interest formula

Instead of diverging, which seems like the most intuitive result, the rate appears to asymptotically approach a constant value approximately equal to 2.718. This is the constant , and is defined by the limit . We can now use this definition to find a formula for continuous compound interest: We now introduce a change of variables; let . We have as , so the limit is still approaching infinity. With substitution of the definition of , we have the continuous compound interest formula: .

## Natural exponential

The exponential base has the interesting property that it is its own derivative, . Try in the graph below:

Find

Check all that apply
Let's prove this statement. Again, let's introduce a change of variables. Let (this is not the -substitution from calculus; we are just using as a placeholder), then . As ,, so we have Let . Then as , . We have Because is a continuous function on its domain, we may move the limit inside and substitute the definition of :