Continuous Compound Interest Calculator

The continuous compound interest calculator finds the future value of money when interest is compounded continuously. Enter the initial balance, nominal annual interest rate, and time in years. The calculator applies the exponential formula, then reports the ending balance, interest earned, growth factor, annual rate, and initial balance.

Continuous compounding is a distinct interest model, not just another name for daily compounding. Monthly compounding credits interest 12 times per year; daily compounding credits it 365 times per year; continuous compounding treats the crediting interval as shrinking all the way to zero. In practical consumer accounts, APY or EAR is usually more common. In finance theory, valuation models, and some advanced rate conversions, continuous compounding is a compact way to describe smooth exponential growth.

This page is focused on the formula A equals P times e raised to r times t. For ordinary periodic compounding with annual, monthly, or daily schedules, use the compound interest calculator. If you know the starting and ending balances and need to solve the rate, use the compound interest rate calculator. For the broader present-value and future-value framework, use the time value of money calculator.

How to use the calculator

Enter the initial balance as the amount invested or saved today. Enter the annual interest rate as a percentage; type 5 for 5%, not 0.05. Enter time in years. Decimals are accepted, so 18 months can be entered as 1.5 years.

The primary result is the balance after the selected number of years. The result panel also shows interest earned, the growth factor, the annual rate, and the original balance. Interest earned can be negative if the annual rate is negative. A principal of zero is allowed and returns a zero future value, but a positive principal is more useful for interpreting the growth factor.

Formula

The continuous compounding formula is:

$A = P \times e^{r \times t}$

The interest earned is:

$\text{interest earned} = A - P$

where A is the future value, P is the initial balance, r is the nominal annual rate written as a decimal, t is time in years, and e is the natural exponential constant. The calculator converts your percentage input into a decimal before applying the formula:

$r = \frac{\text{annual rate as a percent}}{100}$

The growth factor shown in the result is:

$\text{growth factor} = e^{r \times t}$

Because the same growth factor multiplies the principal, doubling the initial balance doubles the future value and the interest earned while leaving the factor unchanged.

Worked example

Use the calculator defaults: an initial balance of $10,000, an annual interest rate of 5%, and a time period of 10 years. The calculator converts 5% to 0.05, multiplies by 10 years, and evaluates the exponential growth factor.

$\text{growth factor} = e^{0.05 \times 10} = e^{0.5}$

That factor is about 1.6487. Multiplying by the $10,000 principal gives a future value of $16,487.21. The interest earned is the difference between the future value and the starting balance, or $6,487.21. These values match the result panel: balance after 10 years, interest earned, growth factor, annual rate, and initial balance.

If the same nominal 5% rate were compounded monthly instead, the future value would be slightly lower because interest would be credited 120 times rather than continuously. The gap is not large for this example, but it grows with higher rates, longer terms, and larger balances.

Nominal rate, effective rate, and continuous growth

The rate entered here is nominal. It is the annual rate used inside the exponent, not the effective annual yield. With continuous compounding, the one-year effective annual rate is higher than the nominal rate whenever the nominal rate is positive:

$\text{effective annual rate} = e^r - 1$

For a 5% nominal continuous rate, the one-year effective rate is about 5.13%. That is the annual return you would compare against APY, EAR, or AER quotes. The effective interest rate calculator can convert continuous and periodic quotes to the same effective annual basis, while the equivalent rate calculator converts between compounding schedules.

This distinction also helps prevent a common mistake: comparing a continuous nominal rate directly with a monthly-compounded APY. The continuous nominal rate describes the input to the exponential formula. The effective annual rate describes the actual one-year increase. They answer different questions, even when they come from the same underlying growth curve.

Practical tips

• Use years for time. For nine months, enter 0.75.
• Enter the rate as a percentage in the form, but use the decimal version when checking the formula by hand.
• Use continuous compounding only when it matches the quote, model, or assignment. Daily compounding is close, but not identical.
• Compare offers using effective annual rates, not nominal rates with different compounding assumptions.
• Do not add deposits or withdrawals to this calculation. It assumes the principal grows without additional cash flows.
• For inflation-adjusted thinking, use a rate that reflects the question you are asking: nominal investment return, real return, or discount rate.

Sources

• Regulation DD Appendix A—Annual Percentage Yield Calculation — eCFR current through 2026-07-09; Authoritative periodic-interest/APY assumptions; algebraic future-value, inverse-rate, and continuous-growth extensions remain disclosed publisher mathematics.
• Calculation scope: The equations and assumptions described above are applied only to values entered in the form. No live rates, prices, tax rules, lender terms, or accounting classifications are fetched. Results are user scenarios, not quotes or prescribed classifications.