Continuous Compounding: formula
Continuous Compounding: worked example
| Starting amount | $10,000 |
|---|---|
| Annual rate | 5.0% |
| Years | 10 |
| Ending balance | $16,487 |
| Interest earned | $6,487 |
| Effective annual rate | 5.1% |
The exponential model holds the continuous rate constant and has no intervening deposits or withdrawals. A quoted APY already includes compounding; entering it as this rate applies a different growth assumption. The difference from a discrete schedule depends on principal, rate and elapsed time, so use the account's actual convention.
Continuous Compounding: changing Annual rate
| Annual rate | Ending balance | Interest earned | Effective annual rate |
|---|---|---|---|
| 3.0% | $13,499 | $3,499 | 3.0% |
| 4.0% | $14,918 | $4,918 | 4.1% |
| 5.0% (example) | $16,487 | $6,487 | 5.1% |
| 6.0% | $18,221 | $8,221 | 6.2% |
| 7.0% | $20,138 | $10,138 | 7.3% |
Continuous Compounding: compare input ranges
| Input | Tested from | To | Ending balance: low input | Ending balance: high input | Swing |
|---|---|---|---|---|---|
| Starting amount | $9,000 | $11,000 | $14,838 | $18,136 | $3,297 (20%) |
| Annual rate | 4.0% | 6.0% | $14,918 | $18,221 | $3,303 (20%) |
| Years | 9 | 11 | $15,683 | $17,333 | $1,649 (10%) |
Continuous Compounding: individual input comparisons
| Input | Value | Ending balance | Interest earned | Effective annual rate |
|---|---|---|---|---|
| Starting amount | $8,000 | $13,190 | $5,190 | 5.1% |
| $9,000 | $14,838 | $5,838 | 5.1% | |
| $10,000 | $16,487 | $6,487 | 5.1% | |
| $11,000 | $18,136 | $7,136 | 5.1% | |
| $12,000 | $19,785 | $7,785 | 5.1% | |
| Annual rate | 3.0% | $13,499 | $3,499 | 3.0% |
| 4.0% | $14,918 | $4,918 | 4.1% | |
| 5.0% | $16,487 | $6,487 | 5.1% | |
| 6.0% | $18,221 | $8,221 | 6.2% | |
| 7.0% | $20,138 | $10,138 | 7.3% | |
| Years | 8 | $14,918 | $4,918 | 5.1% |
| 9 | $15,683 | $5,683 | 5.1% | |
| 10 | $16,487 | $6,487 | 5.1% | |
| 11 | $17,333 | $7,333 | 5.1% | |
| 12 | $18,221 | $8,221 | 5.1% |
Continuous Compounding: combined input changes
| Annual rate / Starting amount | $8,000 | $10,000 | $12,000 |
|---|---|---|---|
| 3.0% | $10,799 | $13,499 | $16,198 |
| 4.0% | $11,935 | $14,918 | $17,902 |
| 5.0% | $13,190 | $16,487 | $19,785 |
| 6.0% | $14,577 | $18,221 | $21,865 |
| 7.0% | $16,110 | $20,138 | $24,165 |
Continuous Compounding: which assumptions move the result?
Continuous Compounding: values and definitions
| Quantity and definition | Value used |
|---|---|
| Starting amount | $10,000 |
| Annual rate | 5.0% |
| Years | 10 |
| Ending balance | $16,487 |
| Interest earned | $6,487 |
| Effective annual rate | 5.1% |
Continuous Compounding: half and double Starting amount
| Starting amount | Ending balance |
|---|---|
| $5,000 | $8,244 |
| $10,000 | $16,487 |
| $20,000 | $32,974 |
Enter the initial principal, an annual continuously compounded rate in percent and elapsed years. A 5% rate is entered as 5; the equation converts it to 0.05. This model has no recurring cash-flow input.
A nominal rate quoted under discrete compounding requires a convention conversion before it represents the same growth assumption.
Common error: Expecting a bank to offer it. Accounts compound daily or monthly, so continuous compounding is a mathematical ceiling and a tool in finance models rather than a product.
Investing & Returns calculators: Compound Interest & Future Value Calculator, Investment Return Calculator, CAGR and Required Annual Return Calculator, Present Value Calculator, Dividend Yield Calculator, IRR Calculator.
Continuous compounding is the limit as interest is added more and more often. The balance is the start times e raised to the rate times the years. $10,000 at 5 percent for 10 years becomes $16,487.21, which is $6,487.21 of interest, and the effective yearly rate is 5.127 percent.
Compare it with monthly and daily compounding at the same rate. The gap is small: monthly compounding ends about $17 lower on this example and daily compounding less than $1 lower, which shows why the frequency matters less than the rate.
For real accounts use the compound interest calculator, and for the effective rate of a quoted APR use the APY calculator. The rule of 72 and future value pages give practical shortcuts. In finance courses this formula links to present value with continuous discounting.
What is the effective annual rate here?
It is e raised to the nominal rate minus 1. At a 5% nominal rate it is 5.13%, the annual growth you would see.
Why is the number e used?
It is the base for continuous growth, arising as the limit of compounding more and more often. It is about 2.71828.
Does any bank compound continuously?
Continuous compounding is the mathematical limit as compounding frequency increases. An actual account follows its disclosed interest and crediting convention. Compare the effective annual result under that convention rather than assuming a continuously compounded model describes the account.