About the Continuous Compounding Calculator
This continuous compound interest calculator applies the formula A = Pe^rt, where interest is compounded an infinite number of times per year. Enter a principal, an annual nominal rate and a time in years (decimals are fine) to get the future value, the interest earned, the equivalent effective annual rate and how long it takes the money to double.
Continuous compounding is the theoretical upper limit of compounding, so it is a staple of finance, economics and calculus classes, and it is the convention used in options pricing models such as Black-Scholes and in many growth and decay problems. Students can use it to check homework; savers can use the comparison table to see how little extra continuous compounding adds over daily or monthly compounding at realistic rates.
The comparison assumes the same nominal annual rate compounded yearly, quarterly, monthly and daily. Enter the rate as a percentage (5 for 5%). For continuous decay problems, or to discount a future amount back to today, the same formula applies with a negative exponent: A = Pe^(−rt).
With the default inputs, the future value (a) is $16,487.21. Change any value above to recalculate instantly.
How to use the continuous compounding calculator
- 1Enter the principal P — the starting amount.
- 2Enter the annual nominal interest rate r as a percentage.
- 3Enter the time t in years; fractions such as 2.5 are allowed.
- 4Read the future value A and the interest earned.
- 5Use the comparison table to see how continuous compounding compares with other frequencies.
Formula and method
Ordinary compound interest is A = P(1 + r/n)^(nt), where n is the number of compounding periods per year. As n grows without limit, (1 + r/n)^n approaches e^r, where e ≈ 2.71828 is Euler's number. That limit gives the continuous compounding formula A = Pe^rt.
The effective annual rate e^r − 1 is the yearly growth that continuous compounding produces, and it is the most any compounding frequency can achieve for a given nominal rate. The doubling time comes from solving 2 = e^rt for t, giving ln 2 ÷ r — the exact version of the "Rule of 70". The comparison table shows the same nominal rate compounded at common frequencies.
- A
- Future value
- P
- Principal (starting amount)
- r
- Annual nominal rate as a decimal
- t
- Time in years
- e
- Euler's number, about 2.71828
Worked examples
$10,000 at 5% for 10 years
A = 10,000 × e^(0.05 × 10) = 10,000 × e^0.5 = $16,487.21. Compounded monthly the same rate gives $16,470.09, so continuous compounding adds only about $17. The effective annual rate is e^0.05 − 1 = 5.1271%.
$5,000 at 8% for 20 years
A = 5,000 × e^1.6 = $24,765.16. At 8% continuous, money doubles every ln 2 ÷ 0.08 ≈ 8.66 years, and the effective annual rate is 8.3287%.
Textbook problem: $1,000 at 3.5% for 2.5 years
rt = 0.035 × 2.5 = 0.0875, and e^0.0875 ≈ 1.091442, so the account grows to $1,091.44 with $91.44 of interest.
Frequently asked questions
What is the formula for continuous compound interest?+
A = Pe^rt, where P is the principal, r the annual rate as a decimal, t the time in years and e ≈ 2.71828. For example $1,000 at 6% for 5 years gives 1,000 × e^0.3 ≈ $1,349.86.
Is continuous compounding much better than daily compounding?+
No. At normal interest rates the difference is tiny. At 5% for 10 years, $10,000 grows to $16,486.65 compounded daily and $16,487.21 continuously — a difference of well under a dollar.
Do banks use continuous compounding?+
Very rarely. Most savings accounts compound daily or monthly and quote an APY. Continuous compounding is mainly used in finance theory, derivatives pricing, and economics and science models of growth and decay.
How do I convert a continuous rate to an APY?+
Use APY = e^r − 1. A 5% continuously compounded rate equals an APY of about 5.127%. To go the other way, the continuous rate equals ln(1 + APY).
How do I solve for time or rate in A = Pe^rt?+
Take natural logs: t = ln(A/P) ÷ r and r = ln(A/P) ÷ t. For instance, to triple money at 5% continuously takes ln 3 ÷ 0.05 ≈ 21.97 years.