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Continuous Compounding Calculator

See the future value when interest compounds continuously.

How it is calculated

The future value is A = P × e(r/100 × t), where P is the starting amount, r is the nominal annual rate as a percentage (5 means 5%), t is the time in years, and e ≈ 2.71828. Interest earned is A − P. The effective annual rate is (er/100 − 1) × 100, the single yearly percentage that would give the same growth with ordinary yearly compounding; it is always slightly higher than the stated rate when r is positive. Principal and years cannot be negative, but the rate can be, which models continuous decay. Time can be fractional, such as 0.5 for six months. Amounts are shown in dollars; the formula itself works for any currency.

Worked example

Put $10,000 at a 5% rate compounded continuously for 3 years. The exponent is 0.05 × 3 = 0.15, and e0.15 ≈ 1.16183. Future value = 10,000 × 1.16183 = $11,618.34. Interest earned = 11,618.34 − 10,000 = $1,618.34. The effective annual rate is e0.05 − 1 = 5.13%. For comparison, yearly compounding at 5% would give $11,576.25 and monthly compounding about $11,614.72, so continuous compounding is the highest of the three.

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Common mistakes

  • Entering the rate as a decimal. Type 5 for 5%; typing 0.05 treats it as 0.05% and the growth almost vanishes.
  • Using months for the time. The exponent needs years, so 18 months is 1.5, not 18, which would give a result far too large.
  • Comparing this directly with a bank's quoted APY. The tool takes the nominal rate r, so a rate that already includes compounding (APY) overstates growth if entered here; convert with ln(1 + APY) first.
  • Expecting a big gain over daily compounding. At typical rates the difference between daily and continuous is tiny; the formula mainly matters in theory and pricing models.
  • Ignoring the sign on a negative rate. A negative rate shrinks the amount, and the interest line then shows a negative number.

Sources

About this calculator

Continuous compounding is the theoretical limit of compound interest as the number of compounding periods per year approaches infinity — instead of compounding monthly, daily, or even hourly, interest is compounded at every instant. It is calculated with the formula A = Pe^(rt), where P is the principal, r is the annual interest rate, t is time in years, and e is Euler's number (approximately 2.71828).

While few real bank accounts actually compound continuously, the formula is foundational in academic finance and derivatives pricing — it underlies the Black-Scholes option pricing model and is used as a clean mathematical benchmark for the maximum possible growth from a given rate. Principal, annual rate and years go into A = P × e^(rt), and the result is the balance you would reach if interest compounded every instant.

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Comments (1)

  • Editorial team

    Doubling time under continuous compounding is the natural log of 2 divided by the rate, about 0.693 over r. At 5 percent that is roughly 13.86 years, and entering 13.86 years here returns almost exactly twice the principal. Handy for checking an exponent before trusting a larger model. What are you modelling, an investment, a population or a decay, and over what period?

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