Continuous Compounding

What Is Continuous Compounding?

Continuous compounding is the mathematical limit that compound interest can reach assuming it's calculated and reinvested into an account's balance over a hypothetically infinite number of periods. While this is absurd 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 month to month, quarterly, or semiannual basis.

Formula and Calculation of Continuous Compounding

Rather than computing interest on a finite number of periods, for example, yearly or month to month, continuous compounding works out interest expecting steady compounding over an infinite number of periods. The formula for compound interest over finite periods of time considers four factors:

PV = the current value of the investment
I = the [stated](/stated-yearly interest-rate) 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)}

Computing the limit of this formula as n approaches vastness (per the definition of continuous compounding) brings about the formula for continuously compounded interest:

FV = PV x e (I x t), where e is the mathematical steady approximated as 2.7183.

Everything Continuous Compounding Can Say to You

In theory, continuously compounded interest means that an account balance is continually earning interest, as well as refeeding that interest once again into the balance so it, too, procures interest.

Continuous compounding computes interest under the assumption that interest will accumulate over an infinite number of periods. Albeit continuous compounding is an essential concept, it's unrealistic in reality to have an infinite number of periods for interest to be calculated and paid. Subsequently, interest is regularly compounded in light of a fixed term, like month to month, quarterly, or every year.

Even with exceptionally large investment sums, the difference in the total interest earned through continuous compounding isn't extremely high when contrasted with traditional compounding periods.

Illustration of How to Use Continuous Compounding

For instance, expect a $10,000 investment procures 15% interest throughout the next year. The accompanying models show the ending value of the investment when the interest is compounded yearly, semiannually, quarterly, month to month, daily, and continuously.

Yearly 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
Month to month 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, a marginal difference.

Highlights

The formula to process continuously compounded interest considers four factors.
Most interest is compounded on a semiannually, quarterly, or month to month basis.
The concept of continuously compounded interest is important in finance even however it's unrealistic in practice.
Continuously compounded interest expects interest is compounded and added once again into the balance an infinite number of times.