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