Loading the player...

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.

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. 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.

RELATED TERMS
  1. Compounding

    Compounding is the process in which an asset's earnings, from ...
  2. Periodic Interest Rate

    The periodic interest rate is the interest rate charged on a ...
  3. Annual Percentage Yield - APY

    The annual percentage yield (APY) is the effective annual rate ...
  4. Stated Annual Interest Rate

    A stated annual interest rate is the return on an investment ...
  5. Future Value - FV

    Future value (FV) is the value of a current asset at a date to ...
  6. Front Fee

    The option premium paid by an investor upon the initial purchase ...
Related Articles
  1. Investing

    Overcoming Compounding's Dark Side

    Understanding how money is made and lost over time can help you improve your returns.
  2. Investing

    The Effective Annual Interest Rate

    The effective annual interest rate is a way of restating the annual interest rate so that it takes into account the effects of compounding.
  3. Personal Finance

    APR and APY: Why Your Bank Hopes You Can't Tell The Difference

    Do you know the difference between Annual Percentage Rate and Annual Percentage Yield? Check out how they can affect your own account balance.
  4. Retirement

    Using Compounding to Boost Retirement Savings

    Allowing growth on your investments to compound over time gives you immense returns when saving for retirement.
  5. Investing

    Compounding: My Favorite Term

    Vanguard CEO Bill McNabb shares why "compounding" is his favorite financial term.
  6. Investing

    Let the Power of Compounding Increase Your Investments

    The power of compounding can exponentially increase the value of your investments over time.
  7. Investing

    How to Make the Time Value of Money Work for You

    How to make the time value of money and power of compounding work for you. Here's a hint: Start saving now.
  8. Investing

    Investing $100 a month in stocks for 30 years

    Find out how you could potentially earn hundreds of thousands of dollars just by investing $100 a month in stocks during your working years.
RELATED FAQS
  1. What formula calculates interest on interest?

    Find out about compounding interest, what it measures, and how to calculate the amount of compound interest accrued using ... Read Answer >>
  2. How to calculate compound loan interest in Excel?

    Find out about compound interest and how to use the compounding interest formula in Microsoft Excel to calculate the compound ... Read Answer >>
  3. How do I calculate compound interest using Excel?

    Learn how to calculate compound interest using three different techniques in Microsoft Excel. Read Answer >>
  4. Yield vs Interest Rate

    Yield is the dividend or interest investors receive from a security, while interest rates are figures charged by a lender, ... Read Answer >>
Hot Definitions
  1. Gross Margin

    A company's total sales revenue minus its cost of goods sold, divided by the total sales revenue, expressed as a percentage. ...
  2. Inflation

    Inflation is the rate at which prices for goods and services is rising and the worth of currency is dropping.
  3. Discount Rate

    Discount rate is the interest rate charged to commercial banks and other depository institutions for loans received from ...
  4. Economies of Scale

    Economies of scale refer to reduced costs per unit that arise from increased total output of a product. For example, a larger ...
  5. Quick Ratio

    The quick ratio measures a company’s ability to meet its short-term obligations with its most liquid assets.
  6. Leverage

    Leverage results from using borrowed capital as a source of funding when investing to expand the firm's asset base and generate ...
Trading Center