Continuous Compound Interest Calculator
Enter your principal, annual interest rate, and time period to find the final balance using continuous compounding, the mathematical limit of compound interest. You can also reverse-solve for the principal, the rate needed, or the time required. Switch the solve mode to explore any unknown. Results update instantly as you type.
Formula
Worked example
You invest $5,000 at a continuously compounded rate of 6% for 5 years. Exponent: 0.06 x 5 = 0.30. Factor: e^0.30 = 1.34986. Final balance: $5,000 x 1.34986 = $6,749.29. Interest earned: $1,749.29. Effective annual rate: e^0.06 - 1 = 6.1837%. Doubling time: ln(2) / 0.06 = 11.55 years.
What is continuous compound interest?
Compound interest means earning interest on your interest as well as your original principal. In practice, banks and savings accounts compound at fixed intervals, such as daily, monthly, or annually. Continuous compounding takes this idea to its mathematical limit: interest is added at every instant, infinitely many times per year. The result is a smooth exponential curve described by the formula A = Pe^(rt), where P is the principal, r is the annual interest rate (as a decimal), t is the time in years, and e is Euler's number, approximately 2.71828. Because e^(rt) grows faster than any finite compounding schedule, continuous compounding always produces a slightly higher balance than monthly or daily compounding at the same nominal rate. The difference is small in practice but meaningful over long periods or at high rates, and understanding it is useful for comparing financial products quoted on different compounding bases.
How to solve for any variable
The formula A = Pe^(rt) contains four variables. This calculator can solve for any one of them when you know the other three. To find the final amount, enter principal, rate, and time. To find the required principal (present value), enter the target final amount, rate, and time; the formula rearranges to P = A / e^(rt). To find the rate you need to reach a target balance, enter principal, final amount, and time; solving gives r = ln(A / P) / t. To find how long it takes, enter principal, final amount, and rate; solving gives t = ln(A / P) / r. All four modes work instantly in this calculator: just change the Solve for dropdown and fill in the three known values.
Effective annual rate and doubling time
The nominal rate r in A = Pe^(rt) is the continuously compounded rate. The equivalent annually compounded rate, called the effective annual rate (EAR), is e^r - 1. For example, a continuous rate of 6% produces an EAR of 6.1837%, which is the rate a standard savings account would need to match the same growth. A related benchmark is the doubling time: how many years until your balance is twice the principal. For continuous compounding, this is ln(2) / r, which simplifies to about 69.3 / r when r is expressed as a percentage, a close relative of the well-known Rule of 72 used for annual compounding. At 6% continuously, the doubling time is ln(2) / 0.06 = 11.55 years.
Continuous compounding vs. monthly compounding
For the same nominal rate, continuous compounding always earns slightly more than monthly compounding. The gap is real but usually modest. A $10,000 deposit at 6% grows to $10,618.37 continuously after one year, versus $10,616.78 compounded monthly, a difference of just $1.59. Over 30 years, the same $10,000 grows to $60,496 continuously versus $60,226 monthly, a gap of $270. The advantage is proportional to the rate and the time period, so it becomes meaningful in high-rate or very long-term scenarios. Most real financial products compound daily or monthly rather than continuously, so the formula is primarily useful for understanding theoretical growth limits, comparing quoted rates, and working backwards to find an implied rate or required time.
Continuous vs. discrete compounding comparison
| Period | Continuous (USD) | Monthly (USD) | Advantage (USD) |
|---|---|---|---|
| 1 year | 10,618.37 | 10,616.78 | 1.59 |
| 5 years | 13,498.59 | 13,488.50 | 10.09 |
| 10 years | 18,221.19 | 18,193.97 | 27.22 |
| 20 years | 33,201.17 | 33,102.04 | 99.13 |
| 30 years | 60,496.47 | 60,225.75 | 270.72 |
Growth of $10,000 at 6% over various periods. Continuous uses A = Pe^(rt); monthly uses A = P(1 + r/12)^(12t).
Frequently asked questions
What is the continuous compound interest formula?
A = Pe^(rt), where A is the final amount, P is the principal, e is Euler's number (approximately 2.71828), r is the annual interest rate expressed as a decimal (so 6% becomes 0.06), and t is the time in years. To solve for any other variable: P = A / e^(rt), r = ln(A / P) / t, and t = ln(A / P) / r.
How does continuous compounding differ from daily compounding?
Both are very close, but continuous compounding is the theoretical limit. Daily compounding applies interest 365 times a year using the formula A = P(1 + r/365)^(365t). Continuous compounding applies it infinitely many times using A = Pe^(rt). In practice the difference is tiny: at 6% over ten years, $10,000 grows to $18,221.19 continuously versus $18,220.44 daily, a gap of about 75 cents.
What is the effective annual rate for a continuously compounded account?
The effective annual rate (EAR) is e^r - 1, where r is the continuously compounded nominal rate. At a nominal rate of 6%, the EAR is e^0.06 - 1 = 6.1837%. This is the annually compounded rate you would need to match the same growth, and it is the figure to use when comparing a continuously compounded product with one that compounds once a year.
How long does it take to double money with continuous compounding?
The doubling time is ln(2) / r, where r is the annual rate as a decimal. Since ln(2) = 0.6931, this is approximately 69.3 divided by the rate as a percentage. At 6% per year, the doubling time is 0.6931 / 0.06 = 11.55 years. This is a continuous-compounding version of the Rule of 72, which divides 72 by the rate for a quick estimate under annual compounding.
Do banks actually use continuous compounding?
Very rarely for consumer products. Banks typically compound daily or monthly, which is close to but not identical to continuous compounding. Continuous compounding is common in financial mathematics, derivatives pricing (notably the Black-Scholes model), and in comparing rates across different compounding bases. When a bank quotes a continuously compounded rate, it is often in a professional or institutional context, not a retail savings account.
Can I use this calculator for negative interest or decay?
Yes. The formula A = Pe^(rt) works for any real value of r. A negative rate models exponential decay, such as the depreciation of an asset or radioactive decay. Enter a negative rate (for example -3 for -3%) and the calculator shows how the balance shrinks over time. The doubling time output becomes a half-life when r is negative.