## Definition

Valuation of financial instruments and projects assumes that interest is usually compounded at discrete intervals, for example, annually, semiannually, quarterly, or monthly. In some cases, though, interest can be added continuously to calculate the future value – a process called continuous compounding.

## Formula

To calculate the future value at continuously compounded interest, use the formula below.

**FV = PV × e ^{rt}**

Here PV is the present value, r is the annual interest rate, t is the number of years, and e is Euler’s number equal to 2.71828.

## Example

Someone has invested $100,000 at a 12% annual fixed interest rate for 10 years. Let’s look at three possible scenarios when interest is added annually, semiannually, and continuously.

Using the formula of discrete compounding and the formula above, we get the following results:

FV _{Annual Compounding} = $10,000 × (1 + 0.12)^{10} = $31,058.48

FV _{Semiannual Compounding} = $10,000 × (1 + 0.12÷2)^{10×2}^{ }= $32,071.35

FV _{Continuous Compounding} = $10,000 × 2.71828^{0.12×10}

Annual compounded interest will return a balance of $31,058.48 after 10 years, semiannually will return $32,071.25, and continuously will be $33,201.17.

## Continuous Compounding vs. Discrete Compounding

Comparison of discrete and continuous compounding is shown in the figure below.

Let’s assume that $1 is invested at an interest rate of 10% for 20 years. During the early years, there is only a slight difference in future values, e.g., if interest is compounded discretely, the future value of $1 will be $1.61 at the end of the fifth year. In contrast, if interest is added continuously, the future value of $1 after 5 years will be $1.65. So the difference of $0.04 is not so great as it may seem. A longer period of time, though, always means a greater difference. For example, it will be $0.66 ($7.39-$6.73) at the end of the twentieth year, which is 66% of the initial $1 invested.

Two drivers make the difference in future values regardless of whether discrete or continuous compounding is applied: interest rate and length of time. The higher interest rate and/or the longer the length of time, the greater the difference in future values between discrete and continuous compounding.