What this calculator does
Continuous compounding is the theoretical limit of compound interest as the number of compounding periods per year grows without bound: instead of compounding annually, monthly or even daily, interest is compounded at every instant. The formula collapses to A = P·e^(rt), where e is Euler's number, roughly 2.71828, principal P grows at annual rate r for t years.
In practice almost no real account compounds continuously, but the formula shows up constantly in finance theory, options pricing and anywhere continuous growth is being modelled. It is also a useful upper bound: no amount of increasing the compounding frequency on a given nominal rate can push the future value past what continuous compounding produces.
The formula
Convert the annual rate to a decimal, multiply it by the number of years to get rt, then raise e to that power and multiply by the principal. The result is compared here against the same principal and rate compounded annually and compounded monthly, to show how close ordinary compounding already gets to the continuous limit.
| Term | Meaning |
|---|---|
| e | Euler's number, approximately 2.71828, the base of natural growth used in continuous compounding. |
| Principal (P) | The starting amount being invested or grown. |
| Rate (r) | The annual interest or growth rate, entered as a percentage and used as a decimal in the formula. |
| Continuous compounding | Compounding applied at every instant rather than at fixed intervals such as monthly or annually. |
The inputs explained
| Field | What to enter |
|---|---|
| Principal ($) | The starting principal or lump sum being grown. |
| Annual interest rate (%) | The annual interest or growth rate, as a percentage. |
| Years | The number of years the amount grows for. |
When to use it
Comparing compounding frequency in theory
Setting the same principal, rate and time side by side under annual, monthly and continuous compounding shows how much of the theoretical maximum an ordinary annual or monthly account is already capturing.
Working through a finance or economics problem
Continuous compounding turns up regularly in textbook growth and option-pricing problems specifically because A = P·e^(rt) is easy to differentiate and integrate, unlike the discrete compounding formula.
Estimating an upper bound on growth
Given a nominal annual rate, continuous compounding shows the highest future value that rate could ever produce, no matter how frequently a real product compounded it.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How future value grows over time under continuous compounding
A $10,000 principal, compounded continuously at 6% per year, over a range of time periods.
| Years | Future value (continuous) |
|---|---|
| 1 yr | $10,618.37 |
| 5 yr | $13,498.59 |
| 10 yr | $18,221.19 |
| 20 yr | $33,201.17 |
| 30 yr | $60,496.47 |
| 40 yr | $110,231.76 |
How future value changes with the annual rate
The same $10,000 principal, compounded continuously for 20 years, at a range of annual rates.
| Annual rate | Future value (continuous) |
|---|---|
| 2% | $14,918.25 |
| 4% | $22,255.41 |
| 6% | $33,201.17 |
| 8% | $49,530.32 |
| 10% | $73,890.56 |
| 12% | $110,231.76 |
Questions
How is continuous compounding different from ordinary compound interest?
Ordinary compound interest applies at fixed intervals, such as once a year or once a month, using A = P(1+r/n)^(nt). Continuous compounding is the limit as those intervals shrink toward zero, giving the simpler A = P·e^(rt), which always produces a slightly higher result for the same nominal rate.
Does any real bank account actually compound continuously?
Essentially none. Continuous compounding is mostly a theoretical convenience in finance and economics. Daily compounding, the most frequent common in practice, already lands extremely close to the continuous figure for typical rates and terms.
Why use e in the formula at all?
e arises naturally as the limit of (1+1/n)^n as n grows without bound, which is exactly what happens to compounding as the number of periods per year increases indefinitely. It is not an arbitrary choice; it is what continuous growth converges to.
Can this formula be used for continuous decline, such as depreciation?
Yes, using a negative rate. A = P·e^(rt) with r negative models continuous decay in the same way it models continuous growth with a positive r, though this calculator is set up for the growth case.
For ordinary compound interest with a selectable compounding frequency, see the compound interest calculator.