Answer first
What this calculator tells you
Calculate the balance when interest compounds continuously, with the effective annual rate. See the upper limit of what compounding can add at a given rate. Formula: A = P × e^(r × t); interest = A − P; effective annual rate = e^r − 1. At the worked-example inputs, the ending balance is $16,487. Holding every other input steady, moving annual rate from 3.0% to 7.0% moves the result from $13,499 to $20,138.
Transparent method
The formula
See the upper limit of what compounding can add at a given rate.
Worked example
Example inputs
How to interpret the result
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.
At the worked-example inputs the ending balance is $16,487. It rises with starting amount, annual rate and years.
Investment returns are uncertain; taxes, fees, volatility and cash-flow timing can materially change results.
Before you rely on it
What to check
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.
The common error
Where people go wrong with continuous compounding calculator
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.
Sensitivity evidence
How annual rate changes the ending balance
Holding every other input at the worked-example value, moving annual rate from 3.0% to 7.0% moves the ending balance from $13,499 to $20,138: a spread of $6,639, or 40% of the worked-example result.
| 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%worked example | $16,487 | $6,487 | 5.1% |
| 6.0% | $18,221 | $8,221 | 6.2% |
| 7.0% | $20,138 | $10,138 | 7.3% |
Every input, tested
Which input moves the ending balance most
Of the 3 inputs, starting amount moves the ending balance most ($3,297 across the range tested) and years moves it least ($1,649).
| Input | Tested from | To | Ending balance at each end | Swing |
|---|---|---|---|---|
| Starting amount | $9,000 | $11,000 | $14,838 to $18,136 | $3,297 (20%) |
| Annual rate | 4.0% | 6.0% | $14,918 to $18,221 | $3,303 (20%) |
| Years | 9 | 11 | $15,683 to $17,333 | $1,649 (10%) |
Two variables at once
Ending balance by annual rate and starting amount
Across the grid the ending balance runs from $10,799 to $24,165. Moving annual rate from 3.0% to 7.0% shifts it by $6,639 at the middle column, and moving starting amount from $8,000 to $12,000 shifts it by $6,595 at the middle row, so annual rate is the bigger lever here.
| 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 |
The highlighted cell is the worked example: $16,487.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Starting amount | $10,000 | Enter the starting amount used in this calculation. |
| Annual rate | 5.0% | Enter the annual rate used in this calculation. |
| Years | 10 | Enter the years used in this calculation. |
| Ending balance | $16,487 | |
| Interest earned | $6,487 | |
| Effective annual rate | 5.1% | |
Inputs, definitions and assumptions
Starting amount
Enter the starting amount used in this calculation. The prefilled worked-example value is $10,000.
Annual rate
Enter the annual rate used in this calculation. The prefilled worked-example value is 5.0%.
Years
Enter the years used in this calculation. The prefilled worked-example value is 10.
How to use this calculator
- 1Verify the inputs. Gather starting amount, annual rate and years from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the ending balance at $16,487. Store your own version of it as Scenario A.
- 3Test one change. Start with starting amount, the input with the biggest effect here: moving starting amount from $9,000 to $11,000 takes the ending balance from $14,838 to $18,136, a swing of 20% of the worked-example figure.
- 4Check the extremes. At half the example starting amount ($5,000) the ending balance is $8,244; at double ($20,000) it is $32,974.
People also ask
Frequently asked questions
How do you calculate continuous compounding?
A = P × e^(r × t); interest = A − P; effective annual rate = e^r − 1. Enter starting amount in dollars and annual rate in percent (5 means 5%). At the worked-example inputs the ending balance is $16,487.
What does the continuous compounding result mean?
See the upper limit of what compounding can add at a given rate. At the worked-example inputs the ending balance is $16,487. It rises with starting amount, annual rate and years.
How much does annual rate change the ending balance?
Holding every other input at the worked-example value, moving annual rate from 3.0% to 7.0% moves the ending balance from $13,499 to $20,138, a spread of $6,639.
What are the limits of this continuous compounding calculator?
Investment returns are uncertain; taxes, fees, volatility and cash-flow timing can materially change results. The tables on this page test annual rate only from 3.0% to 7.0%; a value outside that range is not tabulated here.
Which input moves the ending balance most in the continuous compounding calculator?
Ranked by how far each moves the ending balance across the range tested: starting amount ($3,297, 20%), annual rate ($3,303, 20%) and years ($1,649, 10%).
If I double starting amount in the continuous compounding calculator, does the ending balance double?
Doubling it from $10,000 to $20,000 takes the ending balance from $16,487 to $32,974, which is 2.00 times the worked-example figure. So the result scales almost exactly in proportion. Halving it to $5,000 gives $8,244.
How much does starting amount matter in the continuous compounding calculator?
The worked example uses $10,000. Holding every other input at its worked-example value, moving starting amount from $9,000 to $11,000 takes the ending balance from $14,838 to $18,136, a swing of 20% of the worked-example figure.
How much does years matter in the continuous compounding calculator?
The worked example uses 10. Holding every other input at its worked-example value, moving years from 9 to 11 takes the ending balance from $15,683 to $17,333, a swing of 10% of the worked-example figure.
Which inputs change the interest earned in the continuous compounding calculator?
At the worked-example inputs it is $6,487. Starting amount takes it from $5,838 to $7,136, annual rate takes it from $4,918 to $8,221 and years takes it from $5,683 to $7,333.
Which inputs change the effective annual rate in the continuous compounding calculator?
At the worked-example inputs it is 5.1%. Annual rate takes it from 4.1% to 6.2%.
Sources and evidence
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