What Is Continuous Compounding?
Continuous compounding is the mathematical limit that compound interest can reach ifit'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.
Formula and Calculation of 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. 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 constantapproximated as 2.7183.
Key Takeaways
- Most interest is compounded on a semiannually, quarterly, or monthly basis.
- Continuously compounded interest assumes interest is compounded and added back into the balance an infinite number of times.
- The formula to compute continuously compounded interest takes into account four variables.
- The concept of continuously compounded interest is important in finance even though it’s not possible in practice.
What Continuous Compounding Can Tell You
In theory, continuously compounded interest means that an account balance is constantly earning interest, as well as refeeding that interest back into the balance so that it, too, earns interest.
Continuous compounding calculates interest under the assumption that interest will be compounding over an infinite number of periods.Although continuous compounding is an essential concept, it's not possible in the real world to have an infinite number of periods for interest to be calculated and paid. As a result, interest is typically compounded based on a fixed term, such as monthly, quarterly, or annually.
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.
Example of How to Use Continuous Compounding
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, a marginal difference.
What is compound interest?
Compound interest is interest earned on interest you've received. When interest compounds, each subsequent interest payment will get larger because it is calculated using a new, higher balance. More frequent compounding means you'll earn more interest overall.
How does annual percentage yield (APY) relate to continuous compounding?
Annual percentage yield (APY) is the real rate of return on an investment, taking compounding interest into account. The APY of an account with more frequent, or continuous compounding will be higher than the APY of an account that has infrequent compounding, assuming they both have the same interest rate.
What are the most common compounding periods?
Depending on the situation, interest is typically compounded monthly, quarterly, semi-annually, or annually. Some accounts may even offer daily compounding, though compounding more frequently than that is incredibly unusual.
What is discrete compounding?
Discrete compounding is the opposite of continuous compounding. Instead of interest compounding constantly, it compounds at set intervals, such as daily or monthly.
The Bottom Line
Continuous compounding may be a theoretical concept that can't be achieved in reality, but it has real value for savers and investors. It allows savers to see the maximum amount they could earn in interest for a given period and can be useful when comparing to the actual yield of the account.
