Quick answer
With continuous compounding the future value is A = P × e^(rt). $1,000 at 5% for 10 years grows to $1,648.72, earning $648.72.
Enter the principal, annual rate and years. The calculator returns the future value and interest earned.
Related tools: Simple Interest Calculator: Interest earned or owed without compounding. Compound Interest Calculator: Future value with compound interest and regular contributions. APY Calculator: Convert a nominal rate (APR) into the effective annual yield (APY).
How to use the Continuous Compounding Calculator
- Enter principal ($) (0 or more).
- Enter annual rate (%) (0 or more).
- Enter years (0 or more).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Principal | A number, in $ | ≥ 0 |
| Annual rate | A number, in % | ≥ 0 |
| Years | A number | ≥ 0 |
Formula and method
Formula
A = P·e^(rt)
IEEE-754 double precision; money shown to cents, rates to 4 decimals. Results are estimates, not financial advice.
Limit of compounding as the number of periods tends to infinity.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Principal = 1000; Annual rate = 5; Years = 10.
| Result | Value |
|---|---|
| Future value | $1,648.72 |
| Interest earned | $648.72 |
Example 2
Inputs: Principal = 500; Annual rate = 0; Years = 10.
| Result | Value |
|---|---|
| Future value | $500.00 |
| Interest earned | $0.00 |
The first line is the headline result (future value). Below it the continuous Compounding Calculator lists interest earned, which describe the same case from other angles. IEEE-754 double precision; money shown to cents, rates to 4 decimals. Results are estimates, not financial advice.
The limit of compounding
As compounding becomes more frequent, (1 + r/n)^(nt) approaches e^(rt). Annual compounding gives $1,628.89, daily $1,648.66 and continuous $1,648.72, so the difference between daily and continuous is small.
Where it is used
Continuous compounding is common in theoretical finance and option pricing. Bank accounts normally compound daily or monthly. The effective annual rate of continuous 5% is e^0.05 − 1 = 5.127%.
How the result changes with principal
Holding every other input at the first worked example, this table changes only principal. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Principal | Future value | Interest earned |
|---|---|---|
| 500 | $824.36 | $324.36 |
| 1000 | $1,648.72 | $648.72 |
| 2000 | $3,297.44 | $1,297.44 |
| 4000 | $6,594.89 | $2,594.89 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Years: entering "-1" shows "Years must be at least 0." beside that field and no result.
Choosing between the Continuous Compounding Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Simple Interest Calculator](/finance/simple-interest-calculator/) | principal, annual rate and time | Interest |
| [Compound Interest Calculator](/finance/compound-interest-calculator/) | initial amount, annual interest rate and years | Future value |
| [APY Calculator](/finance/apy-calculator/) | nominal annual rate and compounding | APY |
| [Interest Per Day Calculator](/finance/interest-per-day/) | balance, annual rate and day-count basis | Interest per day |
| [Real Interest Rate Calculator](/finance/real-interest-rate-calculator/) | nominal interest rate and inflation rate | Real interest rate |
Background: interest, saving and investing
A nominal rate (APR) does not include the effect of compounding; the annual percentage yield (APY) does: APY = (1 + r/n)^n − 1. A 5% nominal rate compounded annually is 5.000%, semi-annually about 5.063%, quarterly about 5.095% and monthly about 5.116%, so more frequent compounding helps, but with diminishing returns.
The rate you enter in a projection is an assumption, not a promise: real returns vary and can be negative. Fees and taxes are not modelled unless the page says so.
Assumptions and limitations
- Overflow reported
Results are estimates for planning, not financial advice. Lender terms, fees, taxes and rounding rules can change real figures, so confirm important numbers with your lender or a licensed adviser.
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for years to confirm this.
Frequently asked questions
What is e?
A constant, about 2.71828.
Is continuous compounding better than daily?
Only marginally: $1,000 at 5% for 10 years grows to $1,648.72 continuously and $1,648.66 compounded daily.
How do I find the time to double?
Doubling time is ln 2 ÷ r; at 5% continuous that is 0.693 ÷ 0.05 = 13.9 years.
What does the Continuous Compounding Calculator calculate?
Growth with continuously compounded interest. It returns Future value and Interest earned.
What formula does the Continuous Compounding Calculator use?
A = P·e^(rt). Limit of compounding as the number of periods tends to infinity.
What do I need to enter in the continuous Compounding Calculator?
You enter Principal ($), Annual rate (%) and Years.
How is the result rounded or limited?
IEEE-754 double precision; money shown to cents, rates to 4 decimals. Results are estimates, not financial advice. Overflow reported.
What is an example of a continuous Compounding result?
With principal 1000, annual rate 5 and years 10, the future value is $1,648.72.
Can I rely on this calculator for a decision?
Results are estimates for planning, not financial advice. Lender terms, fees, taxes and rounding rules can change real figures, so confirm important numbers with your lender or a licensed adviser. For this calculator in particular: Overflow reported.
Related calculators
- Simple Interest Calculator — Interest earned or owed without compounding.
- Compound Interest Calculator — Future value with compound interest and regular contributions.
- APY Calculator — Convert a nominal rate (APR) into the effective annual yield (APY).
- Interest Per Day Calculator — Daily interest on a balance at an annual rate.
- Real Interest Rate Calculator — Inflation-adjusted interest rate using the exact Fisher relation.
- Auto Loan Calculator — Monthly car payment after down payment and trade-in.
Sources
- Standard time-value-of-money relations (e.g. Brealey, Myers & Allen, Principles of Corporate Finance)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.