# Continuous Compounding

## What is 'Continuous Compounding'

Continuous compounding is the mathematical limit that compound interest can reach. It is an extreme case of compounding since most interest is compounded on a monthly, quarterly or semiannual basis. Hypothetically, with continuous compounding, interest is calculated and added to the account's balance every infinitesimally small instant. While this is not possible in practice, the concept of continuously compounded interest is important in finance.

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## 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 is:

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

Calculating the limit of this formula as n approaches infinity 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.

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, semi-annually, 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. The difference between the two is only \$0.36. Not much more is earned when compounding an infinite amount of times versus compounding just 365 times. Even if the investment amount was increased to \$10 million, the total difference would only amount to \$358.

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