How to use the continuous compound interest calculator
Start by checking the type of rate you intend to enter. A continuously compounded annual rate differs from a yield that already includes a year's growth, even when both are expressed as annual percentages.
- Enter the initial balance. This is the amount available at the start. Keep later contributions separate.
- Enter the continuous annual rate. For an illustrative 6% rate, enter 6 in a percentage field; the formula uses 0.06.
- Set the time horizon. The formula measures time in years. An equal monthly interval corresponds to one-twelfth of a year.
- Enter any monthly contribution. Use zero for a single deposit. Contributions enter as separate payments; they are not spread continuously across the month.
Check the contribution timing when matching a calculation to your own scenario. The worked example below assumes deposits at the end of each month. Money added earlier has longer to grow at a positive rate.
The continuous compound interest formula
For one starting amount with no additional deposits, the formula is:
A = Pe^(rt)
Here, A is the future value, P is the initial principal, r is the continuously compounded annual rate as a decimal, and t is the time in years. The constant e is approximately 2.71828, and e^(rt) is the growth factor applied to the starting amount.
Continuous compounding is the limit of increasingly frequent compounding at a fixed nominal annual rate. In the expression P × (1 + r/n)^(nt), increasing n without bound leads to Pe^(rt). It does not mean an account pays an unlimited amount of interest: for fixed P, r, and t, the result is finite.
Converting to an effective annual yield
To convert a continuous rate into its effective annual equivalent, use:
Effective annual yield = e^r − 1
An illustrative 6% continuous annual rate therefore corresponds to approximately 6.1837% effective annual yield. The reverse conversion is r = ln(1 + y), where y is the effective annual yield as a decimal and ln is the natural logarithm.
These conversions describe equivalent growth over a year. Entering a quoted effective yield directly as a continuous rate would change the assumption and generally overstate growth for a positive yield. Keep the rate type consistent before comparing results from different calculators.
How monthly contributions are modeled
Monthly deposits remain discrete events even though the money grows continuously between them. Each deposit earns growth only for the time remaining after it enters the balance.
For m equal monthly periods, a constant monthly contribution C, and deposits at month-end, let q = e^(r/12). The future value is:
A = P × q^m + C × (q^m − 1) / (q − 1)
The first term describes the initial balance; the second combines all monthly deposits. The final deposit has no time to grow before the ending valuation. Beginning-of-month deposits would multiply the contribution term by q because each payment receives one additional month of growth.
At a zero rate, q equals 1 and the fraction above cannot be evaluated directly. Use A = P + C × m instead: the final value is simply all the money contributed.
Example: $10,000 with monthly contributions
Assume a $10,000 initial balance, $500 deposited at each month's end, an illustrative constant continuous annual rate of 6%, and a 10-year horizon. There are 120 monthly deposits, with each month treated as one-twelfth of a year.
| Result | Amount |
|---|---|
| Initial balance | $10,000.00 |
| Monthly deposits over 120 months | $60,000.00 |
| Total money contributed | $70,000.00 |
| Modeled earnings | $30,227.71 |
| Future value | $100,227.71 |
The calculation retains precision internally and rounds the displayed amounts to cents. Future value includes your own $70,000 as well as growth; it is not all investment profit. Without monthly deposits, the same initial amount, rate, and horizon produce $18,221.19.
Continuous versus daily compounding
Daily compounding adds interest at discrete daily intervals. Continuous compounding uses an exponential growth factor without a finite compounding interval. At the same positive nominal rate, continuous compounding produces slightly more growth than a finite daily schedule.
For the single $10,000 deposit above, daily compounding at 6% nominal annual interest over 3,650 days, using a fixed denominator of 365, produces $18,220.29. The continuous model produces $18,221.19. This comparison uses ten equal 365-day years rather than actual calendar dates.
For real dates and day-count conventions, use the daily compound interest calculator. The general compound interest calculator supports comparisons across compounding frequencies. Use the compound interest formula calculator to solve an unknown variable in a single-deposit periodic-compounding model.
Continuous compounding is not a standard assumption for every savings account. This model holds the rate constant and excludes taxes, fees, and market volatility. The result is not personalized financial advice. Adjust the inputs above to compare another mathematical scenario.