Continuous Compound Calculator

Background.

Compound interest pays interest on earlier interest, and the more often that happens the faster a balance grows. Annual compounding credits it once a year, monthly twelve times, daily 365 times. Continuous compounding is the end of that sequence: interest added at every instant, so the balance never pauses between credits.

The limit is finite, and it brings in the constant e, about 2.71828. As the number of compounding periods n rises, (1 + r/n)^(nt) closes in on e^(rt), which is why the formula here is A = P × e^(rt). Nothing more can be squeezed out by compounding more often, so the continuous figure is a ceiling on what a given nominal rate can earn.

That ceiling is closer than most people expect. $10,000 at 5% for ten years grows to $16,288.95 with annual compounding, $16,470.09 compounded monthly and $16,486.65 compounded daily; continuous compounding adds only 56 cents beyond daily. Frequency matters most when the rate is high or the term is long.

Few savings accounts compound continuously. The formula earns its place elsewhere: option pricing models such as Black–Scholes, bond and derivatives maths, and any process that grows or shrinks smoothly rather than in steps, which is why the same exponential describes radioactive decay.

The rate you enter is treated as a continuously compounded annual rate, held constant for the whole term. Negative rates are accepted and produce smooth decline. There are no deposits, withdrawals, fees or taxes, so the result is the path of one lump sum left untouched.

What is continuous compound calculator?

Continuous compounding is the limiting case of compound interest in which the compounding interval shrinks to zero. A balance P growing at a nominal continuous rate r reaches P × e^(rt) after t years, where e is Euler's number. The equivalent effective annual rate is e^r − 1, so a 5% continuous rate matches 5.127% compounded once a year. The growth figure is the future value minus the principal: all the interest earned, with nothing withdrawn along the way.

How to use this calculator.

1. 01Enter the starting amount as Principal, in dollars. It is treated as a single lump sum; regular contributions are not modelled.
2. 02Enter the Annual rate as a percentage, for example 5 for 5%. If you have an APY or effective yield instead, convert it first with ln(1 + APY): a 5% APY corresponds to about 4.879% continuous.
3. 03Enter the Time in years. Fractions work, so 18 months is 1.5.
4. 04Read the continuously compounded value and, beneath it, the growth over principal. A negative rate is allowed and shows how much of the principal survives steady decay.

The formula.

A = P × e^(r × t). P is the principal, r the annual rate written as a decimal (5% becomes 0.05), t the time in years and e ≈ 2.71828 the base of natural logarithms. The growth figure is A − P.

The formula comes from the ordinary compound-interest expression P × (1 + r/n)^(n × t), where n is the number of compounding periods per year. Letting n grow without bound sends (1 + r/n)^n to e^r, and the continuous version follows.

Only the product r × t matters. Five per cent for ten years and ten per cent for five years both give an exponent of 0.5 and the same $16,487.21 on $10,000. Because time sits in the exponent, its effect accelerates: at 5% money doubles every ln 2 ÷ 0.05 ≈ 13.86 years, so 30 years multiplies it about 4.48 times rather than three. Over long horizons the rate is the input that moves the answer most, since each extra percentage point compounds through the entire term.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Principles of Finance, 2022. Chapters 6–10. Retrieved 2026-08-06.
2. [2]U.S. Internal Revenue Service. Tax guidance current 2026. Official forms, instructions and tax topics. Retrieved 2026-08-06.
3. [3]U.S. Securities and Exchange Commission. Investor.gov current 2026. Investment products and risk. Retrieved 2026-08-06.