Continuous Compound Interest Calculator
Grow a principal with the continuous-compounding formula A = Pe^rt. Enter a dollar amount, an annual rate in percent, and time in years — the tool also shows how continuous stacks up against plain annual compounding.
Example: with Principal ($) 10000 · Annual rate (%) 5 · Time (years) 10 → Final amount (A): $16,487.21.
- Interest earned$6,487.21 interest earned
- Versus annual compounding$16,288.95 with annual compounding — continuous earns $198.27 more
Computed by the calculator below using its default values. Change any input to see your own numbers.
A = Pe^rt with e ≈ 2.71828. Continuous compounding is the mathematical ceiling — daily compounding lands within pennies of it.
What continuous compounding means
Ordinary compound interest adds interest at fixed checkpoints — yearly, monthly, daily. Continuous compounding is the limit of that process: interest credited at every instant, each sliver immediately earning on itself. The limit has a closed form, A = Pe^rt, where e ≈ 2.71828 is the natural growth constant.
The surprise is how little the last steps add. $10,000 at 5% for 10 years grows to $16,288.95 compounded annually, $16,470.09 monthly, $16,486.65 daily — and $16,487.21 continuously. Going from annual to monthly earns real money; going from daily to continuous earns 56 cents.
Where the formula shows up
No consumer bank credits interest continuously, but the formula is everywhere in finance math: option-pricing models like Black-Scholes quote rates continuously, bond analysts use continuously compounded (log) returns because they add cleanly across periods, and A = Pe^rt is the standard growth model in calculus courses. To convert, a continuous rate r is equivalent to an effective annual rate of e^r − 1 — 5% continuous equals 5.13% effective.
How it’s calculated
A = P × e^(r×t), with r the annual nominal rate as a decimal (the calculator takes percent and divides by 100) and t in years; e ≈ 2.718281828. Interest = A − P. The comparison line uses annual compounding, P × (1 + r)^t. Results are pre-tax and assume no deposits, withdrawals, or fees along the way.
Continuous compounding is a theoretical ceiling, not a bank product — use it for math and modeling, and an APY-based tool for actual account quotes.
$10,000 at 5% for 10 years, by compounding frequency
| Compounding | Final amount | Gain over annual |
|---|---|---|
| Annual | $16,288.95 | — |
| Quarterly | $16,436.19 | $147.24 |
| Monthly | $16,470.09 | $181.14 |
| Daily (365) | $16,486.65 | $197.70 |
| Continuous | $16,487.21 | $198.27 |
Computed with P(1 + r/n)^(nt) and Pe^rt; rounded to cents.
Common mistakes
- Entering the rate as a decimal — this tool takes 5 for 5%; typing 0.05 models a 0.05% rate.
- Using months for t: the rate is annual, so 18 months is t = 1.5, not 18.
- Expecting continuous to beat daily compounding by a lot — the difference is pennies; the big jumps happen between annual and monthly.
- Comparing a continuous rate to a bank APY without converting: effective annual rate = e^r − 1.
Frequently asked questions
What is the continuous compound interest formula?
A = Pe^rt: principal times e raised to rate × time, with the rate as a decimal and time in years. $10,000 at 5% for 10 years is 10,000 × e^0.5 = $16,487.21.
What is e?
Euler's number, about 2.71828 — the natural growth constant. It's what (1 + 1/n)^n approaches as n grows, which is exactly why it appears when compounding happens infinitely often.
How different is continuous from daily compounding?
Barely different. On the default example, daily compounding gives $16,486.65 and continuous gives $16,487.21 — 56 cents apart after a decade on $10,000. Continuous is best understood as the ceiling that daily compounding nearly touches.
How do I convert a continuous rate to an effective annual rate?
EAR = e^r − 1. A 5% continuously compounded rate equals e^0.05 − 1 = 5.127% effective annual — that's the number to compare against a bank's APY.
Does any account actually compound continuously?
Essentially none — banks credit daily or monthly. The formula's home turf is theory and pricing: log returns, option models, and textbook growth problems.