Continuous Compound Interest Calculator — A = Pe^(rt)
Continuous compounding is the mathematical limit of compounding — interest is credited at every instant rather than monthly or yearly. Use this calculator to find the final balance, interest earned, effective annual yield and doubling time for any principal, rate and term.
$
%
years
Principal plus continuously compounded interest
- 1
Rate as decimal
5% ÷ 100 = 0.05 - 2
Exponent (r × t)
0.05 × 10 = 0.5 - 3
e^(r×t)
e^(0.5) = 1.648721Euler's number e ≈ 2.71828 raised to the exponent. - 4
Final amount
10,000 × 1.648721 = 16,487.21
How does this calculator work?
Continuous compounding uses A = P × e^(r × t). At P = $10,000, r = 5 %, t = 10 years: A = 10,000 × e^0.5 ≈ $16,487. The effective annual yield is e^0.05 − 1 ≈ 5.127 %, and the doubling time is ln(2)/0.05 ≈ 13.86 years.
Formula
How this is calculated
Standard compound interest is credited n times per year. As n increases without bound (daily → hourly → every millisecond), the formula converges to A = P × e^(r × t), where e ≈ 2.71828 is Euler's number. At 5 % for 10 years, continuous compounding yields about 0.13 % more than monthly compounding — the gap grows with higher rates and longer terms.
The effective annual yield (APY) is e^r − 1. This lets you compare continuously compounded rates with traditionally quoted rates on the same basis — a 5 % continuously compounded rate is equivalent to a 5.127 % annually compounded rate. The doubling time (where A = 2P) is ln(2) ÷ r ≈ 0.693 ÷ r.
Continuous compounding is used in theoretical finance (Black–Scholes options pricing, bond duration) and some high-frequency savings instruments. In practice, most banks compound daily, which is very close to but not identical to continuous compounding.
Frequently asked questions
At a 5 % nominal rate over 10 years, daily compounding gives A = P × (1 + 0.05/365)^3650 ≈ 1.6487 P, while continuous gives e^(0.05×10) ≈ 1.6487 P — they are nearly identical. The difference only becomes visible at very high rates or very long horizons.
Continuous compounding makes calculus tractable: derivatives of e^(rt) are clean and the log-normal model for asset prices (used in Black–Scholes) relies on continuously compounded returns. It is a mathematical convenience, not a literal bank practice.
APY = e^r − 1. For example, a continuously compounded rate of 6 % equals an annual effective rate of e^0.06 − 1 ≈ 6.184 %. Conversely, to find the continuous rate equivalent to a given APY, use r = ln(1 + APY).
TG we-Calculate Editorial Team. (2026). Continuous Compound Interest Calculator — A = Pe^(rt) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/continuous-compound-calculator
TG we-Calculate Editorial Team. "Continuous Compound Interest Calculator — A = Pe^(rt)." TG we-Calculate. 2026. https://we-calculate.com/calculator/continuous-compound-calculator.
TG we-Calculate Editorial Team, "Continuous Compound Interest Calculator — A = Pe^(rt)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/continuous-compound-calculator
@misc{wecalculate_continuous_compound_calculator, title = {Continuous Compound Interest Calculator — A = Pe^(rt)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/continuous-compound-calculator}}, year = {2026}, note = {TG we-Calculate} }
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