Continuously Compounded Interest: Formula and Worked Example

Continuously compounded interest is calculated with A = P × e^rt. It models growth as the time between compounding events becomes arbitrarily small. For example, $10,000 at a fixed 5% annual rate compounded continuously grows to $16,487.21 after ten years, assuming no additional deposits, withdrawals, fees or taxes.

The word “continuously” does not mean the balance becomes infinite. It describes a mathematical limit. To use the formula correctly, check that the quoted rate is a continuously compounded annual rate, then enter it as a decimal and measure time in years.

The Continuous Compound Interest Formula

A = P × e^rt

Symbol	Meaning	Example
A	Final balance, including the original principal	The amount to calculate
P	Starting principal	$10,000
e	Euler's number, a constant approximately equal to 2.71828	Use the calculator's e-to-the-power function
r	Stated annual rate for continuous compounding, written as a decimal	5% becomes 0.05
t	Time in years when r is an annual rate	10 years; six months would be 0.5 years

The product r × t is the exponent: calculate e raised to that power, then multiply by P. It is not P × e × r × t. OpenStax's explanation of continuous growth gives this formula and identifies the principal, rate and time variables.

The formula returns the final balance. To calculate interest alone, subtract the principal: Interest = A − P.

Worked Example: $10,000 at 5% for Ten Years

Suppose the rate stays at 5% throughout the ten years, with no deposits or withdrawals after the initial $10,000. We exclude fees, taxes and inflation.

Convert the percentage: r = 5 ÷ 100 = 0.05.
Multiply rate by time: r × t = 0.05 × 10 = 0.5.
Calculate the growth factor: e^0.5 ≈ 1.6487212707.
Multiply by the principal: A = $10,000 × e^0.5 ≈ $16,487.21.
Subtract the original deposit: interest ≈ $16,487.21 − $10,000 = $6,487.21.

Keep the calculator's full precision until the final answer. Replacing e with only 2.72 or rounding the growth factor early can change the result.

For a shorter period, the time unit still matters. Six months is 0.5 years, so the same principal and annual rate give $10,000 × e^{0.05 × 0.5} = $10,253.15, rounded to cents.

Continuous vs Annual, Monthly and Daily Compounding

For periodic compounding, use A = P(1 + r/n)^nt, where n is the number of compounding periods per year. Annual compounding uses n = 1, monthly uses n = 12, and this simplified daily model uses n = 365.

The table holds the starting principal at $10,000, the nominal annual rate at 5% and the duration at ten years. Only the compounding method changes.

Compounding method	Final balance	Interest earned
Annual	$16,288.95	$6,288.95
Monthly	$16,470.09	$6,470.09
Daily, using 365 periods per year	$16,486.65	$6,486.65
Continuous	$16,487.21	$6,487.21

In this example, continuous compounding produces about $198 more than annual compounding over ten years. Its advantage over monthly compounding is $17.12, and its advantage over daily compounding is about $0.56. More frequent compounding adds progressively smaller gains at the same nominal rate.

Daily compounding is therefore close to continuous compounding here, but the formulas are different. For a closer look at two common account schedules, see our daily vs monthly compounding comparison.

These figures are model calculations, not forecasts or current account offers. A variable savings rate may change, and investment returns do not normally follow a constant guaranteed rate.

Why Continuous Compounding Does Not Produce Infinite Interest

Increasing n gives interest more opportunities to earn interest, but it also makes the rate for each individual period smaller: that rate is r/n. The combination approaches a finite growth factor, e^rt, for a fixed finite rate and time.

The University of Nebraska–Lincoln's continuous-growth lesson explains this transition from the periodic formula to the continuous formula. You do not need to divide a year into seconds or invent a very large compounding frequency. Use e^rt directly when the problem specifies continuous compounding.

An actual account's daily interest calculation or monthly crediting schedule should still be modeled using its disclosed terms. Continuous compounding is a useful mathematical benchmark, not a substitute for those terms.

Continuous Interest Rate vs APY

A stated 5% continuous annual rate and a 5% effective annual yield represent different growth factors. For a constant rate with interest retained, the effective annual yield, expressed as a decimal, is:

E = e^r − 1

With r = 0.05, E = e^0.05 − 1 ≈ 0.0512710964. That is an effective annual yield of approximately 5.1271%. The University of Northern Iowa's continuous-compounding notes give this effective-yield relationship.

To work in the other direction, use r = ln(1 + E), where ln is the natural logarithm. If the effective annual yield is 5%, the equivalent continuous annual rate is ln(1.05) ≈ 0.0487901642, or 4.8790%. Entering 0.05 directly into e^rt would model a different yield.

For the fixed-rate model, this conversion gives the same growth: P × e^{ln(1 + E) × t} = P(1 + E)^t. Once the effective annual yield is held constant, changing the mathematical representation does not create extra annual earnings.

For US deposit-account disclosures, the CFPB's APY calculation guidance says APY reflects the interest rate and compounding frequency under defined assumptions. Compare the disclosed APY and account terms, including any variable-rate or payout conditions. A nominal-rate comparison alone does not establish which real account is better.

How to Calculate Continuous Interest Yourself

On a scientific calculator, enter the rate as a decimal, multiply it by the time in years, apply the e^x function, and multiply by the starting balance.

In a spreadsheet, the EXP function calculates e raised to a power. The ten-year example is:

=10000*EXP(0.05*10)

This returns approximately 16487.212707. Subtract 10000 for interest alone. If the principal is in A1, the rate entered as 5% is in A2, and the years are in A3, use:

=A1*EXP(A2*A3)

The plain formula assumes one starting deposit. Adding monthly savings requires valuing each deposit for its own remaining investment period. For that situation, use our separate guide to compound interest with monthly contributions.

How Long Does Money Take to Double?

With continuous compounding and a fixed positive rate, set A = 2P and solve 2 = e^rt. Taking natural logarithms gives:

Doubling time = ln(2) ÷ r

At a 5% continuous annual rate, ln(2) ÷ 0.05 ≈ 13.86 years. This is exact for the stated model before rounding. The familiar Rule of 72 is a rough shortcut; the logarithm formula is the direct calculation when continuous compounding is specified.

Can ToolNimbly Calculate Continuous Compounding?

ToolNimbly's compound interest calculator does not have a continuous-compounding setting. It offers daily, monthly, quarterly, semiannual and annual compounding, plus recurring contributions. Use it to compare those periodic compounding schedules; use the formula or spreadsheet example above for the exact continuous model.

To reproduce the periodic rows in this article, enter a starting amount of $10,000, a 5% nominal annual rate, ten years and zero recurring contributions. Change only the compounding frequency. Daily is a close comparison in this example, but it is not the continuous result.

Common Calculation Mistakes

Entering 5 instead of 0.05: in the formula, 5 means 500%. A calculator field labeled as a percentage may instead expect 5; follow its input label.
Using months as years: with an annual rate, six months is 0.5 years.
Treating APY as the continuous rate: convert an effective annual yield with ln(1 + E) before using the continuous formula.
Confusing balance with interest: A includes the starting principal; interest is A − P.
Using one lump-sum formula for later deposits: each contribution earns interest only after it enters the account.