Compound Interest Formula With e: Continuous Growth

The compound interest formula with e is A = Pe^rt, which models continuous compounding: the balance equals the starting principal multiplied by e raised to the annual interest rate times the number of years. Enter the principal, annual rate, and time below to calculate the ending balance and the interest earned.

Formula

For continuous compounding, use A = Pe^rt. The same relationship can be written A = P × exp(r × t), where exp(x) means e raised to the power x.

A is the ending balance, in dollars.
P is the initial principal, in dollars.
e is Euler’s number, approximately 2.71828.
r is the annual interest rate as a decimal, not a percentage. For an input of 5%, use 0.05.
t is the time in years.

The calculator accepts the annual rate as a percentage and converts it by dividing by 100. Its second result is interest earned, calculated as A − P. The result assumes the entered rate remains constant for the full period.

Worked Example

Using the calculator’s defaults—$10,000 principal, 5% annual interest, and 10 years—convert the rate to a decimal: r = 5 ÷ 100 = 0.05. Then apply the continuous-compounding formula:

Calculate the exponent: r × t = 0.05 × 10 = 0.5.
Evaluate the exponential factor: e^0.5 ≈ 1.648721.
Multiply by the principal: A = $10,000 × 1.648721 ≈ $16,487.21.
Subtract the starting principal: $16,487.21 − $10,000 = $6,487.21 in interest.

The displayed results are an ending balance of $16,487.21 and interest earned of $6,487.21. Values are rounded to cents for display; calculations use the unrounded exponential result until the output.

Assumptions

The annual rate is fixed throughout the entire period.
Interest is compounded continuously, rather than at monthly or other discrete intervals.
The entered time is measured in years; fractions of a year are permitted.
No deposits, withdrawals, fees, taxes, or changes in rate are included.

How Continuous Compounding Works

With ordinary compounding, interest is added at set intervals. For example, annual compounding adds interest once per year, while monthly compounding adds it monthly. Continuous compounding is the mathematical limit approached as compounding intervals become increasingly frequent. The constant e appears because the balance grows in proportion to its current size at every instant.

For a nominal annual rate r, a discrete-compounding formula is A = P(1 + r/n)^nt, where n is the number of compounding periods per year. As n increases without bound, this expression approaches Pe^rt. Continuous compounding is therefore a useful model and comparison point, but it does not mean that a real account necessarily credits interest continuously.

For instance, at 5% for 10 years, the continuous model gives about $16,487.21 on $10,000. Monthly compounding at the same nominal annual rate uses $10,000(1 + 0.05/12)¹²⁰, which is about $16,470.09. The difference is about $17.12. The comparison holds the nominal rate and time constant; actual products may use different rate definitions or crediting rules.

When to Use This Formula

Use Pe^rt when a problem explicitly specifies continuous compounding, when studying exponential growth, or when comparing a continuous model with another compounding convention. It is also useful for understanding how principal, rate, and time affect a modelled balance.

Increasing the principal scales the result in the same proportion.
Increasing the rate raises the exponential growth factor.
Increasing time gives the balance more time to grow; the effect is not a fixed dollar addition each year.

This formula is not a general-purpose way to calculate every savings balance or investment return. Check the stated compounding frequency and rate definition first. If interest is compounded quarterly, monthly, or daily, use the formula for that schedule rather than assuming continuous compounding.

Common Errors and Limits

Using 5 instead of 0.05: The formula’s r is a decimal. The calculator converts a percentage input such as 5 into 0.05.
Mixing time units: If the rate is annual, enter time in years. For example, 18 months is 1.5 years.
Confusing interest with the ending balance: A includes the original principal. Interest alone is A − P.
Treating continuous growth as a product feature: The formula describes a compounding model, not a promise that a financial institution credits interest at every instant.
Rounding too early: Rounding the exponential factor or intermediate balance can slightly alter the final cents. Keep full precision until the final display.

The calculation does not account for changing rates, additional contributions, withdrawals, inflation, or product-specific rules. A negative rate can be represented mathematically, but this calculator limits its input to nonnegative rates; its results should not be interpreted as a forecast of future returns.

Frequently Asked Questions

What is the compound interest formula with e?
A = Pe^rt, where P is principal, r is the annual rate as a decimal, and t is time in years. It calculates the balance under continuous compounding.

What does e mean in the formula?
e is Euler’s number, approximately 2.71828. It is the mathematical constant used to describe continuous exponential growth.

Do I enter the interest rate as a percentage or decimal?
This calculator takes a percentage, such as 5 for 5%, and converts it to a decimal internally. In the written formula, use 0.05.

Is continuous compounding better than monthly compounding?
At the same positive nominal annual rate and over the same period, the continuous-compounding formula produces a slightly larger modelled balance than monthly compounding. That mathematical comparison does not establish which real account or product is preferable.

How do I calculate interest earned rather than the total balance?
Calculate A = Pe^rt, then subtract the initial principal: interest = A − P.

This result is an estimate based on the inputs and assumptions; the lender’s disclosure gives the exact figures.

Formula used

in p | Principal | $ | 10000 | 0 | 1000000000
in apr | Annual interest rate | % | 5 | 0.01 | 100
in yrs | Time | years | 10 | 0 | 100
out bal | Ending balance | $ | p * exp((apr / 100) * yrs) | 2
out int | Interest earned | $ | bal - p | 2

Each line is one input (in, sel, list) or one result (out: label, unit, formula, decimals). How the calculators work

Method: the standard loan formula for this calculation, fixed rate and monthly compounding unless stated, rounded as each result states. Not financial advice; no lender, rate or product is recommended. See formulas and assumptions, the disclaimer or report an error.