Investopedia/James Chen: What is Continuous Compounding?

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Continuous Compounding
Reviewed by James Chen
Updated Feb 15, 2019
What is Continuous Compounding

Continuous compounding is the mathematical limit that compound interest can reach if it's calculated and reinvested into an account's balance over a theoretically infinite number of periods. While this is not possible in practice, the concept of continuously compounded interest is important in finance. It is an extreme case of compounding, as most interest is compounded on a monthly, quarterly or semiannual basis.
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Understanding Compound Interest
BREAKING DOWN Continuous Compounding

Instead of calculating interest on a finite number of periods, such as yearly or monthly, continuous compounding calculates interest assuming constant compounding over an infinite number of periods. Even with very large investment amounts, the difference in the total interest earned through continuous compounding is not very high when compared to traditional compounding periods.
Continuous Compounding Formula and Calculation

The formula for compound interest over finite periods of time takes into account four variables:

    PV = the present value of the investment
    i = the stated interest rate
    n = the number of compounding periods
    t = the time in years

The formula for continuous compounding is derived from the formula for the future value of an interest-bearing investment:

Future Value (FV) = PV x [1 + (i / n)](n x t)

Calculating the limit of this formula as n approaches infinity (per the definition of continuous compounding) results in the formula for continuously compounded interest:

FV = PV x e (i x t), where e is the mathematical constant approximated as 2.7183.
An Example of Interest Compounded at Different Intervals

As an example, assume a $10,000 investment earns 15% interest over the next year. The following examples show the ending value of the investment when the interest is compounded annually, semiannually, quarterly, monthly, daily and continuously.

    Annual Compounding: FV = $10,000 x (1 + (15% / 1)) (1 x 1) = $11,500
    Semi-Annual Compounding: FV = $10,000 x (1 + (15% / 2)) (2 x 1) = $11,556.25
    Quarterly Compounding: FV = $10,000 x (1 + (15% / 4)) (4 x 1) = $11,586.50
    Monthly Compounding: FV = $10,000 x (1 + (15% / 12)) (12 x 1) = $11,607.55
    Daily Compounding: FV = $10,000 x (1 + (15% / 365)) (365 x 1) = $11,617.98
    Continuous Compounding: FV = $10,000 x 2.7183 (15% x 1) = $11,618.34

With daily compounding, the total interest earned is $1,617.98, while with continuous compounding the total interest earned is $1,618.34.
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Related Terms
Compounding
Compounding is the process in which an asset's earnings, from either capital gains or interest, are reinvested to generate additional earnings over time.
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Time Value of Money - TVM
The time value of money is the idea that money presently available is worth more than the same amount in the future due to its potential earning capacity.
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Compound Interest
Compound interest is interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan.
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Compound
Compound is the ability of an asset to generate earnings, which are then reinvested to generate still more earnings.
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Future Value (FV)
Future value (FV) is the value of a current asset at a date to come based on an assumed rate of growth over time.
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Stated Annual Interest Rate
A stated annual interest rate is the return on an investment (ROI) that is expressed as a per-year percentage.
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