Continuous Compound Interest Calculator - Future Value

Calculate continuously compounded interest and future value from principal, annual rate, and years using FV = P e^(rt).

Enter principal, the annual rate, and years to project future value when interest compounds continuously at the limit of infinite frequency.

FV = P × e^(r t)

Principal Amount ($)

Annual Interest Rate (%)

Time Period (Years)

About the Continuous Compound Interest Calculator

Continuous compounding is the mathematical limit of compounding interest more and more often. As the number of discrete periods per year grows without bound, the discrete factor (1 + r/n)^(n t) approaches e^(r t), where e is Euler’s number, about 2.71828. The continuous compound interest calculator applies that limit: future value equals principal times e raised to the rate times time. Finance students meet the formula in time-value-of-money chapters. Quantitative roles use it in some derivatives and growth models. Practical savers rarely receive true continuous compounding from a bank, but the formula is the clean upper bound on what a given nominal rate can produce as frequency increases. Enter P as the starting amount, r as an annual percent (5 means 0.05), and t in years. The calculator computes P × exp(r t) and reports interest as future value minus principal. There is no contribution field because the closed form here is for a single lump sum. Compared with monthly compounding at the same nominal rate, continuous compounding is slightly higher. The gap is small at everyday rates and short terms, and it becomes more visible as r t grows. That is why textbooks present continuous compounding as both a limit and a convenient exponential model. Use cases include checking a textbook problem, converting a continuously compounded yield quoted in a model into a dollar future value, and illustrating why APY rises with compounding frequency. If a model quotes a continuously compounded rate of 5%, this page is the right mapping to a ten-year balance. If a savings account quotes a 5% APY with daily compounding, do not treat that APY as r in e^(r t) without converting; APY is already an effective annual yield. A continuously compounded rate r_cc relates to effective annual yield by APY = e^(r_cc) − 1. Caveats matter for anyone leaving the classroom. Bank products almost always compound daily or monthly, not continuously. Taxes, fees, and inflation are omitted. The exponential path assumes a constant rate with no deposits or withdrawals. Negative rates are mathematically allowed in the exponential but are rejected here because required fields must be greater than zero. Very large r t values grow quickly; they are not a forecast of asset prices. Currency is treated as a pure number formatted in dollars for display, so keep the principal in one money unit. When you compare continuous compounding with discrete compounding, hold P, r, and t fixed and only change the compounding convention. The continuous compound interest calculator answers the continuous side of that comparison. For monthly or annual schedules with contributions, use a discrete compound interest calculator instead. Record the rate convention next to the result so a reviewer knows e^(r t) was used. For professional work, confirm whether a quoted yield is continuously compounded, annually compounded, or an APY, because mixing conventions is a common source of mispriced cash flows. The exponential should stay a transparent identity, not a marketing flourish.

Continuous Compound Interest Examples

Each example uses FV = P e^(rt) with the annual rate converted to a decimal.

Inputs	Result	How to read it
$10,000 principal at 5% for 10 years, continuous compounding	Future value $16,487.21	Interest earned is $6,487.21, a bit more than monthly compounding at the same nominal rate.
$5,000 at 7% for 5 years, continuous compounding	Future value $7,095.34	The exponent is 0.35, so the balance grows by a factor of e^0.35.
$25,000 at 4% for 15 years, continuous compounding	Future value $45,552.97	A long horizon at a modest continuous rate still nearly doubles the principal.

How to Use the Continuous Compound Interest Calculator

Enter the principal in dollars.
Enter the annual continuously compounded rate as a percent, such as 5 for 5%.
Enter the time period in years.
Select Calculate to see future value and interest earned under FV = P e^(rt).

Continuous Compound Interest FAQ

What does continuous compounding mean?

It is the limit of compounding infinitely often. The growth factor becomes e raised to the rate times time, which is slightly higher than daily or monthly compounding at the same nominal rate.

Is this how savings accounts actually work?

Almost never. Banks typically compound daily or monthly. Continuous compounding is a modeling convention and an upper bound, useful for theory and for yields that are explicitly quoted as continuously compounded.

How does APY relate to a continuous rate?

Effective annual yield equals e^(r) − 1 when r is the continuously compounded annual rate. Do not paste an APY into the rate field unless you have converted it back to a continuous rate.

Can I include monthly deposits?

Not on this page. The continuous compound interest calculator values a single principal. For recurring contributions, use a discrete compound interest calculator with an explicit payment frequency.

Why is e in the formula?

The number e is the base of natural exponential growth. It appears because (1 + r/n)^(n t) converges to e^(r t) as n goes to infinity, which is the definition of continuous compounding.