Loading the player...

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.

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.

RELATED TERMS
  1. Compounding

    The ability of an asset to generate earnings, which are then ...
  2. Compound Interest

    Compound Interest is interest calculated on the initial principal ...
  3. Discrete Compounding

    Discrete compounding refers to the method by which interest is ...
  4. Effective Annual Interest Rate

    Effective Annual Interest Rate is an investment's annual rate ...
  5. Time Value of Money - TVM

    The idea that money available at the present time is worth more ...
  6. Periodic Interest Rate

    The interest rate charged on a loan or realized on an investment ...
Related Articles
  1. Investing

    Learn Simple And Compound Interest

    Interest is defined as the cost of borrowing money, and depending on how it is calculated, can be classified as simple interest or compound interest.
  2. Investing

    Accelerating Returns With Continuous Compounding

    Investopedia explains the natural log and exponential functions used to calculate this value.
  3. Managing Wealth

    Dissecting the Simple Interest Formula

    Simple interest ignores the effect of compounding: it's only calculated on the principal amount. This makes it easier to calculate than compound interest.
  4. Investing

    Calculating Future Value

    Future value is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today.
  5. 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.
  6. Investing

    How does Compound Interest Work?

    A quick way to understand the impact of compound interest is to ask yourself if you’d rather receive $100,000 a day for a month, or start with a penny on day one and double it every day for those ...
  7. Investing

    Overcoming Compounding's Dark Side

    Understanding how money is made and lost over time can help you improve your returns.
  8. Personal Finance

    How Interest Rates Work on Savings Accounts

    Here's what you need to know to grow your rainy-day fund.
  9. Retirement

    Why Investors Should Care About Compound Interest

    Learn about compounding interest and how it impacts savings decisions, debt management, investment strategies and retirement planning.
  10. Investing

    The Difference Between Compounding Interest and Simple Interest

    Interest is the cost a borrower pays to use someone else’s money. Interest can be either simple or compounded.
RELATED FAQS
  1. What formula can I use to calculate interest on interest?

    Find out more about compounding interest, what it measures and how to calculate the amount of compound interest accrued using ... Read Answer >>
  2. How often is interest compounded?

    Understand what compound interest is and how the compounding of interest applies to the benefit of investors or creditors, ... Read Answer >>
  3. How do I use the rule of 72 to calculate continuous compounding?

    Find out why the rule of 72 does not accurately reflect the growth caused by continuous compounding, and which number can ... Read Answer >>
  4. What is the difference between continuous compounding and discrete compounding?

    Learn to differentiate between and calculate the continuous and discrete compounding formulas for interest-generating investments ... Read Answer >>
  5. How do mutual funds compound interest?

    Learn how mutual funds can grow wealth over time through the magic of compound interest by reinvesting dividends back into ... Read Answer >>
Hot Definitions
  1. Master Of Business Administration - MBA

    A graduate degree achieved at a university or college that provides theoretical and practical training to help graduates ...
  2. Liquidity Event

    An event that allows initial investors in a company to cash out some or all of their ownership shares and is considered an ...
  3. Job Market

    A market in which employers search for employees and employees search for jobs. The job market is not a physical place as ...
  4. Yuppie

    Yuppie is a slang term denoting the market segment of young urban professionals. A yuppie is often characterized by youth, ...
  5. SEC Form 13F

    A filing with the Securities and Exchange Commission (SEC), also known as the Information Required of Institutional Investment ...
  6. Four Percent Rule

    A rule of thumb used to determine the amount of funds to withdraw from a retirement account each year. The four percent rule ...
Trading Center