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Continuous compounding

What is Continuous Compounding?

Continuous compounding is the mathematical limit to which compound interest can reach if it is calculated and reinvested into account balances indefinitely in theory. While this is not possible in practice, the concept of continuous compounding is important in finance. This is an extreme case of compound interest, as most interest is calculated monthly, quarterly, or semiannually.

Formula and calculation of continuous compound interest

Instead of calculating interest over a finite number of periods (such as annually or monthly), continuous compounding calculates interest assuming continuous compounding over an infinite number of periods. The formula for finite time compounding takes into account four variables:

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

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

Final Value (FV) = PV x [1 + (i / n)](Down)

Calculating the limit of this formula as n approaches infinity (according to the definition of continuous compounding) yields the formula for continuous compounding:

FV = PV xe (ixt)where e is a mathematical constant approximately 2.7183.

key takeaways

  • Most interest is compounded semiannually, quarterly, or monthly.
  • Continuous compounding assumes that interest is compounded and added back to the balance indefinitely.
  • The formula for calculating continuous compound interest takes into account four variables.
  • The concept of continuous compounding is important in finance, although impossible in practice.

What continuous compounding can tell you

In theory, continuous compounding means that the account balance is continuously earning interest, and that interest is fed back into the balance so that it earns interest as well.

Continuous compounding calculates interest under the assumption that it will compound over an infinite number of periods. Although continuous compounding is a fundamental concept, in the real world it is impossible to have an infinite number of periods in which interest is calculated and paid. Therefore, interest is usually compounded on a fixed term basis (such as monthly, quarterly, or annually).

Even if the investment amount is very large, the total interest earned through continuous compounding is not very large compared to traditional compounding periods.

Example of how to use continuous compounding

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

  • Annual compound interest: 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)) (4×1) = $11,586.50
  • Compounded monthly: FV = $10,000 x (1 + (15% / 12)) (12×1) = $11,607.55
  • Daily compound interest: FV = $10,000 x (1 + (15% / 365)) (365×1) = $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, and with continuous compounding, the total interest earned is $1,618.34, a small difference.

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