Continuous Compounding Calculator
Continuous growth amount, interest, and multiple.
What it does
Continuous compounding is the limit of splitting the year into ever smaller pieces. Interest earns interest instantly, so growth follows the exponential e^(r t). Banks quote it for theory and some products, while most accounts compound daily or monthly just below it.
The tool above applies the limit. With $1,000 at 5% for 10 years, the factor is 1.64872 and the amount is $1,648.72 with $648.72 interest at 1.65 x. At 7% for 20 years, $5,000 becomes $20,276.00 with $15,276.00 interest. Monthly compounding trails by a small but real gap.
Doubling has a clean form here: years equal ln 2 over r. At 7% that is 9.9 years, near the Rule of 72 estimate of 10.3. Raising rate beats extending time early, while time dominates late because the exponent multiplies both.
This page models one fixed nominal rate with no deposits or withdrawals. It leaves out taxes, fees, and inflation. For account shopping, compare effective annual rates on equal footing.
How to use
- Enter the principal in dollars.
- Enter the nominal annual rate percent.
- Enter whole or fractional years.
- Read the future amount first.
- Read interest, then the multiple.
- Compare against monthly compounding to see the gap.
Formula
- P
- Starting dollars.
- r
- Nominal annual rate as a decimal.
- t
- Years of compounding.
Nominal rate compounded continuously for t years. At zero rate or zero time the amount equals principal. Discrete frequencies need their own (1+r/n)^(n t) form.
Worked example
Example: $1,000 at 5% for 10 years
Defaults: $1,000 at 5% nominal with continuous compounding for 10 years.
- Inputs: $1,000 at 5% for 10 years, continuous.
- Exponent: 0.05 times 10 is 0.5.
- Factor: e to the 0.5 is 1.64872.
- Amount: 1000 times 1.64872 is $1,648.72.
- Interest: $1,648.72 minus $1,000 is $648.72 at 1.65 x. Tool shows the same.
Reference table
| Terms | Amount | Interest |
|---|---|---|
| 1000 at 5% 10yr | $1,648.72 | $648.72 |
| 5000 at 7% 20yr | $20,276.00 | $15,276.00 |
| 2500 at 0% 5yr | $2,500.00 | $0.00 |
| 2000 at 10% 5yr | $3,297.44 | $1,297.44 |
| 100 at 100% 1yr | $271.83 | $171.83 |
Limits
- One fixed rate. Variable rates need segments.
- No cash flows. Deposits need annuity math.
- Before tax and fees.
- Nominal input. Real rate needs inflation split.
Common mistakes
- Using percent as decimal directly, off by 100x.
- Pricing monthly accounts with the continuous form.
- Reading multiple as percent gain.
- Forgetting taxes on the interest.
- Comparing nominal across frequencies as if equal.
Try it yourself
Quick checks. Answers sit tucked under each question.
$5,000 at 7% for 20 years?
Exponent is 1.4, factor 4.0552. Amount is $20,276.00 with $15,276.00 interest at 4.06 x.
$2,500 at 0% for 5 years?
Exponent is 0. At 0% or 0 years the amount is principal: $2,500.00 with $0.00 interest.
Questions
What is the continuous formula?
A = P times e^(r t). Here $1,648.72 on $1,000.
How does this differ from monthly compounding?
Continuous compounds every instant, so it tops any finite frequency at the same nominal rate.
What is the exponent?
Rate / 100 times years. Here 0.5.
What happens at zero?
At zero rate or zero years the amount is principal. Here $5,000 stays $5,000.
How long to double?
Double when e^(r t) is 2, so t is ln 2 / r. At 7% that is 9.9 years.
Is this spendable gain?
No. Before tax, fees, and inflation. Real gain is lower.