# Continuous compound interest

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.

- Page: https://www.acalculator.org/finance/continuous-compound-interest-calculator
- JSON spec: https://www.acalculator.org/finance/continuous-compound-interest-calculator.json
- Version: dc06b6424c82

## Default answer

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.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| p | Starting amount | The deposit or principal at the start. |
| r | Yearly interest rate | The nominal yearly rate, compounded continuously. |
| t | Years | How long the money grows, in years; decimals allowed (1.5 is 18 months). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| amount | Final amount | P × e^(r × t). |
| interest | Interest earned | The final amount minus the starting amount. |
| effective | Effective yearly rate | e^r − 1: the yearly rate that gives the same growth compounded once a year. |
| monthly | With monthly compounding | P × (1 + r/12)^(12t), for comparison. |
| yearly | With yearly compounding | P × (1 + r)^t, for comparison. |

## 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.

## Assumptions

- 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.

## Worked examples

1. p = $1,000.00, r = 10%, t = 1 gives amount = $1,105.17, interest = $105.17, effective = 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. p = $10,000.00, r = 5%, t = 10 gives amount = $16,487.21, interest = $6,487.21, monthly = $16,470.09, yearly = $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. p = $5,000.00, r = 0%, t = 20 gives amount = $5,000.00, interest = $0.00, effective = 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).

## FAQ

### 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.

## Sources

- OpenStax, College Algebra 2e, section 6.1 Exponential Functions: the continuous growth formula A(t) = P·e^(rt) for principal P, rate r and time t; Example 11, $1,000 at a nominal 10% compounded continuously for 1 year is about $1,105.17. Retrieved 2026-10-18. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions