Continuous Compound Interest Calculator
Calculate continuous compound interest using A = Pe^rt, including optional monthly contributions. Compare continuous growth with annual, monthly, and daily compounding.
Growth Over Time
Money Composition Over Time
This chart shows how your balance grows from three sources: your initial principal in the bottom blue band, your monthly contributions in the middle orange band, and interest earned on both in the top reddish-purple band. Over time, the interest portion can grow more quickly as returns compound.
Why 10 Years of Growth Is Not Just Twice 5 Years
It is easy to assume that if an investment gains a certain percentage over 5 years, the gain over 10 years should be about twice as large. Compound growth does not work that way. Returns build on the balance that has already accumulated, so growth over longer periods is not linear.
A simple example
Suppose you invest $10,000 and it grows at a steady 10% per year.
After 5 years, the investment would grow to approximately $16,105.
After 10 years, it would grow to approximately $25,937.
| Time | Balance | Total gain | Cumulative increase |
|---|---|---|---|
| Start | $10,000 | — | 0% |
| 5 years | $16,105 | $6,105 | 61.05% |
| 10 years | $25,937 | $15,937 | 159.37% |
The gain after 5 years is about $6,105, but the gain after 10 years is about $15,937. The 10-year gain is therefore much more than twice the 5-year gain.
This happens because the second 5 years do not start with the original $10,000. They start with the larger balance created during the first 5 years. Future returns are then earned on both the original investment and the growth that has already occurred.
A real-world illustration
Market returns show the same principle, although actual returns vary from year to year. At the time this example was written, a recent S&P 500 market snapshot showed approximately 73.63% growth over 5 years and 255.20% over 10 years.
It would be incorrect to assume that doubling the 5-year return should produce the 10-year return. The two periods include different market conditions, and investment growth compounds over time rather than increasing at a constant linear rate.
Historical market returns are shown only as an illustration of how long-term growth can behave. Past performance does not guarantee future results.
Key idea: Time does more than simply add years to an investment. With compounding, later growth is applied to a larger balance.
Formulas Used
Compound Interest:
Continuous Compounding:
Monthly Contribution Growth: M × (gm − 1) / (g − 1) × gf
Monthly Growth Factor: g = (1 + r/n)n/12 for periodic compounding and g = er/12 for continuous compounding.
Where:
- P = principal
- r = annual interest rate as a decimal
- n = compounding periods per year
- t = time in years
- e = Euler's number, approximately 2.71828
- M = monthly contribution
- g = effective growth factor for one month
- m = completed months
- f = fractional month after the last completed monthly contribution
- A = final amount
Understanding Compound Interest
Compound interest is interest earned on both your original investment and the interest that has already been added to it. Over time, this compounding effect can significantly increase the value of your savings or investments.
Use the calculator above to estimate future growth based on your initial deposit, interest rate, contribution amount, and compounding frequency.
Want to see it used on real numbers? The Compound Interest Guide walks through an emergency fund, a house down payment, and a 25-year retirement scenario step by step, using the exact fields on this page.
How Continuous Compound Interest Works
Continuous compounding represents the theoretical limit of compound interest where interest is added an infinite number of times per year. Instead of compounding annually, monthly, or daily, the investment grows continuously every moment.
The continuous compounding formula is:
For example, $10,000 invested at a 7% annual rate for 10 years grows to $20,137.53 with continuous compounding. The same principal grows to $20,136.18 with daily compounding, a difference of $1.35.
Continuous compounding is mainly a mathematical model. Most savings accounts, loans, and investments use a stated compounding schedule. For a real financial product, select the frequency in its terms and compare quoted annual percentage yields when available.
Why Is There an "e"?
The number e appears because continuous compounding assumes interest is being added every instant of time.
As compounding frequency increases from annual to monthly, daily, hourly, every second, and finally continuous compounding, the standard compound interest formula approaches a limit:
Annual → Monthly → Daily → Hourly → Every Second → Continuous
Continuous compounding is essentially the mathematical limit of increasingly frequent compounding.
Because interest is constantly being added to the balance, continuous compounding produces slightly higher returns than daily compounding at the same annual rate.
What you can calculate
- Future investment value
- Total contributions over time
- Total interest earned
- Growth from recurring monthly deposits
- Effects of different compounding frequencies
- Investment growth scenarios with variable interest rates
Inputs explained
- Principal Amount – your starting balance or initial investment.
- Monthly Contribution – recurring deposits added during the investment period, this can be zero if you want to calculate without contribution.
- Annual Interest Rate– expected yearly return expressed as a percentage. Historical note: the S&P 500 has returned roughly 6.7% per year after inflation since 1957, though actual returns vary widely from year to year.
- Compounding Frequency – how often interest is added to the balance, such as monthly, quarterly, or daily.
- Time Period – total investment duration in years.
- Interest Rate Variance Range – optional high and low growth scenarios used for the chart projection.
Compound interest formula
Standard compound interest is calculated using:
A = P(1 + r/n)^nt
Where:
- A = final amount
- P = principal investment
- r = annual interest rate
- n = compounding periods per year
- t = time in years
Why compounding frequency matters
More frequent compounding generally increases the final investment value because interest is added and reinvested more often. Daily compounding usually produces slightly higher returns than monthly or annual compounding at the same nominal interest rate.
Example calculation
Suppose you invest $10,000 at a 7% annual return compounded monthly, while contributing an additional $200 each month for 20 years. The calculator estimates the final balance, total contributions, and total interest earned while also visualizing growth over time.