# Continuous Compounding Definition and Formula

![Continuous Compounding](https://www.investopedia.com/thmb/ri-MvCb3PAThyqsXe7naXZaLspk=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Continuous-Compounding-FINAL-287562ad21a44099ae9a85d42a28b04e.jpg)

*While continuous compounding is a useful theoretical concept, it's not achievable in practice. — Investopedia / Jiaqi Zhou*

> **Definition**
>
> Continuous compounding is the limit of compound interest when it is calculated and reinvested into an account an infinite number of times.

## What Is Continuous Compounding?

Continuous compounding is the point at which compound interest reaches its maximum potential, being calculated and added to an account's balance without limit. While this is not possible in practice, the concept of continuously compounded interest is important in finance. It is an extreme case of [compounding](https://www.investopedia.com/terms/c/compounding.asp), as most interest is compounded on a monthly, quarterly, or semiannual basis.

### 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.

## Formula and Calculation of Continuous Compounding

Instead of [calculating interest](https://www.investopedia.com/articles/investing/020614/learn-simple-and-compound-interest.asp) 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](https://www.investopedia.com/terms/s/stated-annual-interest-rate.asp) 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 \[1 + (i / n)\]

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 **, where e is the mathematical constant approximated as 2.7183.

## 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](https://www.investopedia.com/terms/c/compoundinterest.asp).

Continuous [compounding](https://www.investopedia.com/terms/c/compound.asp) calculates interest under the assumption that interest will be compounded 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. 

> **Important**
>
> 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))  = $11,500
*   **Semi-Annual Compounding:** FV = $10,000 x (1 + (15% / 2))  = $11,556.25
*   **Quarterly Compounding:** FV = $10,000 x (1 + (15% / 4))  = $11,586.50
*   **Monthly Compounding:** FV = $10,000 x (1 + (15% / 12))  = $11,607.55
*   **Daily Compounding:** FV = $10,000 x (1 + (15% / 365))  = $11,617.98
*   **Continuous Compounding:** FV = $10,000 x 2.7183  = $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.

## Real-World Applications of Continuous Compounding

It's somewhat difficult to find continuous compounding products in financial industries; however, there are some real world applications of continuous compounding. Those applications include:

*   **Options Pricing:** Continuous compounding plays a big part in the [Black-Scholes option pricing model](https://www.investopedia.com/terms/b/blackscholes.asp), one of the most widely used tools in finance. The model assumes that interest accrues continuously over time, which allows for more precise pricing for certain types of options.
*   **Exponential Growth Models:** In fields like economics, continuous compounding is used to model exponential growth and decay. It helps describe systems where change happens at every instant, such as continuously growing investments or continuously depreciating assets.
*   **Financial Engineering and Derivatives:** Many advanced financial instruments rely on models that assume continuous compounding to simplify complex calculations. It allows financial engineers to create and value structured products with embedded features, like path-dependent options.
*   **Discounted Cash Flow Analysis:** Though less common in everyday corporate finance, continuous discounting is sometimes used in theoretical [DCF models](https://www.investopedia.com/terms/d/dcf.asp). This approach assumes that future cash flows are being discounted continuously, offering a more refined present value estimate.

## Limitations of Continuous Compounding

In reality, financial systems do not operate with the kind of precision or immediacy that is the baseline of continuous compounding. The idea of continuously and instantly compounding interest, while mathematically elegant, doesn't align with the operational limitations of real-world finance. For instance, banking platforms simply do not have the infrastructure to continuous compound every deposit across every individual.

This leads to the second major criticism—its inherently theoretical nature. Continuous compounding is not used in day-to-day consumer or commercial banking products. Most commonly used banking items such as CDs, savings accounts, or loans typically use monthly, quarterly, or at most daily compounding. Therefore, aside from the applications mentioned above, continuous compounding really isn't applied to many everyday contexts.

Another limitation is the potential for confusion among those new to finance or unfamiliar with calculus-based models. Because the formula involves exponential functions and a [mathematical constant (e)](https://www.investopedia.com/terms/e/eulers-constant.asp), it's just a more complicated, complex way of calculating interest. The abstract nature of continuous compounding can act as a barrier to financial literacy when simpler, more discrete compounding models might be easier to use with not a ton of implications.

## What Is Compound Interest?

Compound interest is interest earned on the 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 compared to the actual yield of the account.

> **Article Sources**
>
> 1.  Harvard Business School. "[Time Value of Money (TVM): A Primer](https://online.hbs.edu/blog/post/time-value-of-money)."