A = Pe^(rt) · n → ∞

Continuous Compound Interest

Q: What's the continuous compound interest formula?

A = Pe^(rt) where P is principal, r is the annual rate (decimal), t is years, and e ≈ 2.71828. The formula is the limit of (1 + r/n)^(nt) as n → ∞.

The mathematical limit of compounding. We model it with daily compounding (n = 365) — the practical stand-in — and the closed-form Pe^(rt) at zero deposits.

Load scenario$10k · 10y · 7%$5k · 30y · 8%Daily + deposits

Live·Updates on every change

Inputs

Currency

Initial amount

Annual interest rate

Time horizon

Years:

Months:

Compound frequency

Regular contributions: (optional)

Deposit amount: (optional)

Annual deposit increase? (optional)

Inflation & tax⌄

Annual inflation rate

Shown as a "real" value in today's money.

Tax on gains

Applied once at the end of the horizon.

Future value

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After

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Contributed

—

Principal + deposits

Interest earned

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Effective APY

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Growth over time

Principal, contributions, and interest — stacked year by year.

PrincipalContributionsInterest

Year-by-year breakdown

Opening balance, deposits, interest, closing balance.

Show table

Year	Opening	Net flows	Interest	Closing

The formula

Compounding, taken to its limit.

A = P(1 + r/n)^nt is the general compound interest formula. Let n go to infinity and it collapses intoA = Pe^rt — continuous compounding, the upper bound of what any schedule can produce. It's an idealization: no bank actually posts interest 86,400 times a second. But it's the clean form that underpins most theoretical finance math.

APY at 5% nominal

Monthly (n=12)	5.1162%
Daily (n=365)	5.1267%
Continuous (n→∞)	5.1271%

Continuous is the ceiling. Daily is close enough for every practical scenario.

FAQ

Frequently asked

What's the continuous compound interest formula?

A = Pe^rt where P is principal, r is the annual rate (decimal), t is years, and e ≈ 2.71828. The formula is the limit of (1 + r/n)^(nt) as n → ∞.

How close is daily to continuous?

Very close. At 5% nominal, daily compounding gives an APY of 5.1267% and continuous gives 5.1271% — a difference of 0.0004 percentage points. For every practical savings or investment scenario, daily is a stand-in for continuous.

Does any real account compound continuously?

No. Continuous compounding is a mathematical idealization. Banks post interest in discrete chunks — daily is the finest schedule in common use. The continuous formula is useful for closed-form bond and option-pricing math.

How does this calculator model continuous compounding?

It uses daily compounding (n = 365) as the practical stand-in, which lands within cents of the exact A = Pe^rt result: $10,000 at 5% for 10 years grows to $16,486.65 compounded daily versus $16,487.21 continuously. The APY shown is the daily figure, which matches the continuous value at two decimal places.

Why bother with continuous if daily is close enough?

For textbook problems, closed-form math, and theoretical finance — e.g. Black-Scholes option pricing, bond yield-to-maturity derivations — the continuous form is the natural one. For practical savings growth, use any discrete frequency.

The theoretical ceiling, on your screen.

Change the rate and horizon to see how fast the continuous bound pulls away from annual.

Open the calculator Read the formula