acalculator

Continuous compound interest

Type the starting amount, the yearly rate and the number of years. The continuous compound interest calculator gives the final amount from A = Pe^rt, the interest earned, the effective yearly rate, and what monthly or yearly compounding would give instead.

Your numbers

Final amount
$16,487.21

$10,000.00 at 5% compounded continuously for 10 years grows to $16,487.21.

Interest earned
$6,487.21
Effective yearly rate
5.1271%
With monthly compounding
$16,470.09
With yearly compounding
$16,288.95

Final amount: $16,487.21. $10,000.00 at 5% compounded continuously for 10 years grows to $16,487.21.

The results are estimates for information only. They are not financial, tax, or legal advice. Check the numbers with your lender or a qualified professional before you decide. Terms of use

How to calculate

Calculates what a deposit grows to with interest compounded continuously, A = Pe^(rt), with the interest earned, the effective yearly rate, and monthly and yearly compounding for comparison.

Example with the default inputs (Starting amount $10,000.00, Yearly interest rate 5%, Years 10): $10,000.00 at 5% compounded continuously for 10 years grows to $16,487.21.

Method: A = P × e^(r × t), with r the yearly rate as a decimal and t in years. Interest = A − P. Effective yearly rate = e^r − 1.

  • The rate is a nominal yearly rate compounded continuously, and it stays the same for the whole time.
  • No deposits or withdrawals after the start, and no taxes or fees.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Starting amount $1,000.00, Yearly interest rate 10%, Years 1 gives Final amount $1,105.17, Interest earned $105.17, Effective yearly rate 10.517092%.Source: Example 11: $1,000 at 10% compounded continuously for 1 year is about $1,105.17. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions (retrieved 2026-10-18)
  2. Starting amount $10,000.00, Yearly interest rate 5%, Years 10 gives Final amount $16,487.21, Interest earned $6,487.21, With monthly compounding $16,470.09, With yearly compounding $16,288.95.Source: Example 11: $1,000 at 10% compounded continuously for 1 year is about $1,105.17. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions (retrieved 2026-10-18)
  3. Starting amount $5,000.00, Yearly interest rate 0%, Years 20 gives Final amount $5,000.00, Interest earned $0.00, Effective yearly rate 0%.Source: Example 11: $1,000 at 10% compounded continuously for 1 year is about $1,105.17. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions (retrieved 2026-10-18)

How it works

With P the starting amount, r the yearly rate as a decimal (the typed percent ÷ 100), and t the time in years:

  • Final amount A = P × e^(r × t).
  • Interest earned = A − P = P × (e^(r × t) − 1).
  • Effective yearly rate = e^r − 1, shown as a percent with 4 decimal places.
  • With monthly compounding = P × (1 + r ÷ 12)^(12 × t), for comparison.
  • With yearly compounding = P × (1 + r)^t, for comparison.

P runs from $0 to $1 billion, r from 0% to 100%, t from 0 to 100 years (decimals allowed). Money shows to the cent, rounded half up. e^x − 1 is computed directly (expm1) so a small r × t keeps its digits.

Assumptions

  • The rate is a nominal yearly rate compounded continuously, and it does not change.
  • There are no deposits or withdrawals after the start, and no taxes or fees.

Worked examples by hand

$1,000 at 10% for 1 year (OpenStax Example 11). A = 1,000 × e^0.1 = 1,000 × 1.105170918 = $1,105.17. Interest = $105.17. Effective rate = e^0.1 − 1 = 10.5171%.

$10,000 at 5% for 10 years (the default). A = 10,000 × e^0.5 = 10,000 × 1.648721271 = $16,487.21; interest $6,487.21. Monthly: 10,000 × (1 + 0.05 ÷ 12)^120 = $16,470.09. Yearly: 10,000 × 1.05^10 = $16,288.95.

$5,000 at 0% for 20 years. e^0 = 1, so A = $5,000 and the interest is $0.

Other questions people ask

What is the formula for continuous compound interest?

A = P × e^(r × t), where P is the starting amount, r the yearly rate as a decimal, t the time in years, and e ≈ 2.71828. $1,000 at 10% for one year grows to 1,000 × e^0.1 = $1,105.17.

How much more does continuous compounding earn than monthly?

A little. $10,000 at 5% for 10 years grows to $16,487.21 compounded continuously, $16,470.09 monthly and $16,288.95 yearly.

What is the effective yearly rate of continuous compounding?

e^r − 1. At 5% it is e^0.05 − 1 = 5.1271%; at 10% it is 10.5171%. That is the rate that gives the same growth compounded once a year.

Is continuous compounding the limit of compounding more often?

Yes. (1 + r/n)^(n × t) grows toward e^(r × t) as the number of periods n a year grows without end, so continuous compounding is the most any compounding schedule can earn at the same rate.

How long does money take to double with continuous compounding?

ln 2 ÷ r years, about 0.693 ÷ r. At 5% that is 13.86 years.

Do banks compound continuously?

Most savings accounts compound daily or monthly and state the result as an APY. Continuous compounding is used in finance formulas and is the upper limit of what frequent compounding gives.