Continuous compounding

Continuous compounding is the limiting case of compound interest where interest accrues at every instant, so future value equals present value multiplied by e to the power of the rate times time.

ByHoang TruongUpdated July 5, 2026

See it move

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Marta invests €5,000 at a 6% annual rate for four years. Compounded once a year, it grows to €5,000 × 1.06⁴ = €6,312.38. Compounded continuously, using e^(rt), it grows to €5,000 × e^0.24 = €6,356.25 — €43.87 more, because interest accrues at every instant rather than once a year.

Where it fits

TopicTime Value of MoneyAdvancedSubjectCorporate FinanceAdvanced

The formula

LaTeX

FV = PV \times e^{rt}

Variables

$F V$: Future value (€)
$P V$: Present value (€)
$e$: Euler's number, approximately 2.71828 (dimensionless)
$r$: Annual continuously compounded rate (decimal)
$t$: Time (years)

Gives the future value of an amount growing at a continuously compounded annual rate over time t, measured in years.

LaTeX

EAR = e^{r} - 1

Variables

$E A R$: Equivalent effective annual rate (decimal)
$e$: Euler's number, approximately 2.71828 (dimensionless)
$r$: Annual continuously compounded rate (decimal)

Converts a continuously compounded rate into the effective annual rate it is equivalent to under ordinary once-a-year compounding.

Check yourself

PracticeCORE

An investment of €8,000 earns interest at a continuously compounded annual rate of 5%. Using e ≈ 2.71828, what is its value after 3 years?

Select an answer to check your understanding.