Continuous Compound Interest Calculator

Continuous compounding is the idea of interest being added not once a year, not monthly, not even daily, but at every instant. It is the mathematical ceiling that all the other compounding frequencies climb toward but never quite reach. You will meet it in finance courses, in bond pricing, and in any model where growth happens smoothly rather than in steps.

The formula is short: A = P times e raised to the power of (r times t). Here P is the starting principal, r is the annual interest rate written as a decimal, t is the time in years, and e is Euler’s number, a constant of about 2.71828 that shows up whenever something grows in proportion to its own size. So 1,000 dollars at a 5 percent rate for 10 years becomes 1,000 times e^(0.05 times 10) = 1,000 times e^0.5, which is about 1,648.72.

The part worth understanding is how little continuous compounding actually beats daily compounding. Take that same 1,000 dollars at 5 percent for 10 years. Continuous gives 1,648.72. Daily gives 1,648.61. The difference over a decade is eleven cents. Yearly compounding, by contrast, gives 1,628.89, so the gap between yearly and daily is 19.72 dollars, roughly 180 times larger than the gap between daily and continuous.

That is the shape of the whole thing: the jump from yearly to monthly is worth real money, monthly to daily is worth a little, and daily to continuous is a rounding error. Continuous compounding earns its place in finance not because it pays more but because e^(rt) is far easier to differentiate and integrate than a discrete sum, which is why it turns up throughout bond pricing and option theory.

What the continuous case does give you cleanly is the effective annual yield, also called the APY (Annual Percentage Yield): it is simply e^r minus 1. A 5 percent nominal rate compounded continuously yields about 5.13 percent a year.

Enter your principal, the nominal annual rate, and the number of years. The calculator returns the future value, the interest earned, the effective annual yield, and how the continuous figure compares against plain once-a-year compounding so you can see the gap for yourself.