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Continuous Compound Interest Calculator
Calculate the future value of money compounded continuously with the formula A = Pe^rt. See the interest earned, the effective annual rate and doubling time, and compare the result with annual, monthly, daily and hourly compounding.
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Enter your values to see the result.
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How to use the Continuous Compound Interest Calculator
- Enter the principal, the starting amount.
- Enter the annual interest rate in percent (negative for continuous decay).
- Enter the time and choose years, months or days.
- Read the future value, interest and effective annual rate.
- Compare compounding frequencies in the table to see how close daily compounding gets to continuous.
The continuous compounding formula
A = P · ert
It comes from the regular compound interest formula A = P(1 + r/n)nt: as the number of compounding periods n goes to infinity, (1 + r/n)n approaches er. Related formulas:
- Interest earned = P(ert − 1)
- Effective annual rate = er − 1
- Principal needed = A · e−rt
- Rate needed = ln(A ÷ P) ÷ t; time needed = ln(A ÷ P) ÷ r
Effective annual rate by compounding frequency
| Nominal rate | Annual | Monthly | Daily | Continuous |
|---|---|---|---|---|
| 2% | 2.0000% | 2.0184% | 2.0201% | 2.0201% |
| 5% | 5.0000% | 5.1162% | 5.1267% | 5.1271% |
| 8% | 8.0000% | 8.3000% | 8.3278% | 8.3287% |
| 10% | 10.0000% | 10.4713% | 10.5156% | 10.5171% |
| 20% | 20.0000% | 21.9391% | 22.1336% | 22.1403% |
Frequently asked questions
What is the continuous compound interest formula?
A = P × e^(r × t), where P is the principal, r is the annual rate as a decimal, t is the time in years and e ≈ 2.71828 is Euler's number. $10,000 at 5% for 10 years grows to 10,000 × e^0.5 = $16,487.21.
How is continuous compounding different from daily compounding?
Continuous compounding is the limit of compounding more and more often: (1 + r/n)^(nt) approaches e^(rt) as n grows. The difference from daily compounding is tiny; on $10,000 at 5% for 10 years it is only about 56 cents.
What is the effective annual rate with continuous compounding?
EAR = e^r − 1. A 5% nominal rate compounded continuously gives an effective annual rate of about 5.1271%, slightly more than the 5.1267% from daily compounding.
Where is continuous compounding used?
It is mostly used in finance theory, such as option pricing models and continuously compounded returns, and in science for exponential growth and decay of populations, radioactive material or bacteria. Most bank accounts compound daily or monthly.
How long does it take to double with continuous compounding?
Doubling time = ln(2) ÷ r ≈ 0.6931 ÷ r. At 5% that is about 13.86 years. This is where the rule of 69 comes from, a close cousin of the rule of 72.
Can I use a negative rate?
Yes. A negative rate models continuous decay, for example depreciation or radioactive decay, and the result is A = P × e^(−kt).
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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