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.

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.

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.

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.