Continuous Compounding
Value with interest compounded continuously.
Formula
Variables
Example
$1000 at 5% compounded continuously for 3 years: 1000·e^0.15 ≈ $1161.83.
How it works
This formula gives the future value of money when interest is added so frequently that the compounding happens continuously, not just monthly or daily. It comes from the limit of repeatedly compounding over smaller and smaller time intervals, which leads to the exponential function e^{rt}. The exponential growth reflects that each instant, the balance increases by a tiny amount proportional to its current size.
When to use it
Use this when a problem explicitly says interest is compounded continuously, or when you are comparing continuous growth to other compounding methods in finance or science. It also appears in any situation modeled by constant proportional growth, such as population growth or radioactive decay with a sign change.
Worked example, step by step
Invest $2500 at an annual rate of 4.8% compounded continuously for 6 years. A = Pe^{rt} A = 2500e^{(0.048)(6)} A = 2500e^{0.288} A \approx 2500(1.3339) A \approx 3334.75 Final answer: $3334.75
Common mistakes to avoid
- Using 4.8 instead of 0.048 for the interest rate
- Forgetting that t is measured in years, not months or days unless converted
- Rounding too early and losing accuracy in the final amount
Did You Know?
Compounding more and more often approaches a limit set by the number e — continuous compounding.
Share this formula
More in Finance & Interest
View allSimple Interest
BasicInterest earned on principal at a simple rate.
Compound Interest
BasicValue of principal compounded n times per year.
Future Value
BasicFuture worth of a present sum.
Present Value
BasicPresent worth of a future sum.
Future Value of Annuity
AdvancedFuture value of regular equal payments.
Present Value of Annuity
AdvancedPresent value of a series of equal payments.