What is the continuous compounding formula?

FV = PV × ert, where r is the annual rate as a decimal and t the time in years. The present value form is PV = FV × e−rt.

Why does the formula use e?

Because (1 + 1/m)m tends to e as m grows, so compounding infinitely often turns (1 + r/m)mt into ert.

How much more does continuous compounding earn?

Only a little more than daily or monthly compounding. At 6% for 10 years it adds about 14 per 5,000 compared with monthly.

What is the effective annual rate for continuous compounding?

EAR = er − 1. For r = 5% that is 5.127%.

Continuous Compounding Formula (Pe^rt)

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Continuous Compounding Calculator

Continuous compounding is the limit of compounding more and more often: yearly, monthly, daily, every second. As the number of periods per year goes to infinity, (1 + r ÷ m)^mt approaches e^rt, where e ≈ 2.71828. The future value of a deposit is then FV = PV × e^rt, often written A = Pe^rt, and the present value of a future amount is FV × e^−rt.

Formula

FV = PV × e^rt PV = FV × e^−rt

Where

FV: future value
PV: present value (principal)
e: Euler’s number, 2.718281828…
r: annual interest rate, as a decimal
t: time in years

Calculator

Principal (PV)

Annual rate (r) (%)

Time in years (t)

Result and step-by-step solution

Future value (continuous)

9,110.59

Compare: annual compounding gives 8,954.24, monthly gives 9,096.98.

Exponent
r × t = 0.06 × 10 = 0.6
Growth factor
e^0.6 = 1.8221188
Multiply
FV = 5,000.00 × 1.8221188 = 9,110.59
Interest earned
9,110.59 − 5,000.00 = 4,110.59

How to use the formula

In practice few bank accounts compound continuously, but the formula is used throughout finance because it is mathematically convenient: continuously compounded rates add over time, and they are the standard in option pricing models such as Black-Scholes and in many economics and science growth problems.

In the example, 5,000 at 6% for 10 years grows to 9,110.59 with continuous compounding. The calculator compares that with 8,954.24 for annual and 9,096.98 for monthly compounding, which shows that most of the benefit of frequent compounding is already captured at monthly. The equivalent effective annual rate is e^0.06 − 1 = 6.18%, which you can confirm with the EAR calculator.

Frequently asked questions

What is the continuous compounding formula?: FV = PV × e^rt, where r is the annual rate as a decimal and t the time in years. The present value form is PV = FV × e^−rt.
Why does the formula use e?: Because (1 + 1/m)^m tends to e as m grows, so compounding infinitely often turns (1 + r/m)^mt into e^rt.
How much more does continuous compounding earn?: Only a little more than daily or monthly compounding. At 6% for 10 years it adds about 14 per 5,000 compared with monthly.
What is the effective annual rate for continuous compounding?: EAR = e^r − 1. For r = 5% that is 5.127%.

Last reviewed: September 26, 2026. Calculations run in your browser and were checked against numpy-financial, SciPy and published spreadsheet examples.