Interest basics Updated November 2026

Continuous compounding

With continuous compounding, interest is added at every instant and the balance grows as A = P x e^(r x t), where e is 2.71828. It is the upper limit of compounding more and more often. The effective annual rate is e^r - 1, and present value is FV x e^(-r x t).

Syntax

A = P x e^(r x t)   PV = FV x e^(-r x t)

Part	What it is
`P`	Principal
`r`	Continuously compounded annual rate as a decimal
`t`	Time in years

Examples

P = 1,000, r = 5%, t = 3
A = 1,000 x e^(0.05 x 3)

Result: A = 1,161.83

Excel: =1000*EXP(5%*3)

P = 1,000, r = 8%, t = 10
A = 1,000 x e^0.8

Result: A = 2,225.54

Present value of 5,000 in 2 years at 6% continuous
PV = 5,000 x e^(-0.06 x 2)

Result: PV = 4,434.60

Excel: =5000*EXP(-6%*2)

Common mistakes

Using 1 + r in the exponent base instead of e: the formula is e^(rt), not (1 + r)^t.
Expecting continuous compounding to be dramatically larger than daily: 6% continuous gives 6.1837% effective versus 6.1831% daily.
Treating a continuously compounded rate as if it were an effective rate when comparing with a bank quote; convert with e^r - 1 first.

Practice Continuous compounding

These drills check your answer on live data, free, no account needed.

Continuous compoundingA fund grows at 2.25% per year, compounded continuously. What does a $1,000 investment grow to in 15 years?
Effective rate with daily compoundingA short-term business line of credit charges 8.25% APR, compounded daily on a 365-day year. What is the effective annual rate?

Questions

Where is continuous compounding used?

Mainly in finance theory: option pricing (Black-Scholes), bond math and log returns, because continuously compounded rates add across time and make the algebra simpler.

How do I convert an annual effective rate to a continuous rate?

r_continuous = ln(1 + EAR). An effective 5% is ln(1.05) = 4.879% continuously compounded. In Excel: =LN(1+5%).

Effective annual rateThe effective annual rate turns a nominal rate with monthly, quarterly or daily compounding into the true yearly rate: (1 + i/m)^m - 1. Examples and Excel.
Compound interestThe compound interest formula A = P(1 + r/n)^(nt), what each symbol means, examples for yearly and monthly compounding, the Excel FV function and errors.
APR vs APYAPR is the yearly rate without compounding; APY includes it. Convert with APY = (1 + APR/n)^n - 1, worked examples and the Excel EFFECT and NOMINAL functions.
Rule of 72The rule of 72 estimates doubling time as 72 divided by the rate. Examples at 6%, 9% and 18%, how accurate it is, the exact ln(2) formula and Excel NPER.