Continuous compounding is an important concept in mathematics, finance, and investment theory. It describes a situation in which interest is added to an investment continuously rather than at fixed intervals, such as annually, quarterly, monthly, or daily.

In ordinary compound interest, interest is calculated and added to the principal at the end of each compounding period. In continuous compounding, the number of compounding periods approaches infinity, making the process continuous in a mathematical sense.

1. What Is Continuous Compounding?

Continuous compounding is the mathematical process of calculating compound interest at every possible instant.

In compound interest, the interest earned during one period becomes part of the principal for the next period. Consequently, future interest is calculated on both the original principal and the accumulated interest.

As the frequency of compounding increases, the accumulated amount also increases, assuming a positive interest rate and the same nominal annual rate.

For example, interest may be compounded annually, monthly, daily, or continuously. Continuous compounding represents the limiting case in which the number of compounding periods per year approaches infinity.

Suppose the initial principal is $P$, the annual interest rate is $r$, and interest is compounded $n$ times per year.

The interest rate per compounding period is:$$\frac{r}{n}$$​The total number of compounding periods in $t$ years is:

$$nt$$

Using the standard compound interest formula, we get:

Where:

• $P$ is the initial principal.
• $r$ is the nominal annual interest rate expressed as a decimal.
• $n$ is the number of compounding periods per year.
• $t$ is the time in years.
• $A$ is the accumulated amount after $t$ years.

This formula works for monthly, daily, or other fixed-frequency compounding.

Increase the Compounding Frequency

In continuous compounding, interest is compounded infinitely frequently in the mathematical limit. Therefore, we let the number of compounding periods per year approach infinity.

$$n\to \mathrm{\infty }$$

Starting from the compound interest formula, we obtain:

$$A=P\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{r}{n})}^{nt}$$

We can rewrite the expression inside the limit as:

$$A=P\underset{n\to \mathrm{\infty }}{\lim }{\left[{(1+\frac{r}{n})}^{n\mathrm{/}r}\right]}^{rt}$$This works because:$$\frac{n}{r}\times rt=nt$$

From calculus, we know that:

Euler’s number is an irrational mathematical constant with an approximate value of 2.71828. It naturally appears in continuous growth and decay processes.

As $n\to \mathrm{\infty }$, $x\to \mathrm{\infty }$ for a positive interest rate. Therefore,

$$\underset{n\to \mathrm{\infty }}{\lim }{(1+\frac{r}{n})}^{n\mathrm{/}r}=e$$

Substituting this result into our equation gives:

$$A=P[e{]}^{rt}$$

This is the formula for continuous compounding.

Difference Between Compound Interest and Continuous Compounding

The main difference is the frequency at which interest is added to the principal.

In ordinary compound interest, the number of compounding periods per year is finite. In continuous compounding, the compounding frequency approaches infinity.

For a positive nominal annual interest rate, continuous compounding produces a slightly higher accumulated amount than compounding at any finite frequency, provided the principal, time, and nominal rate remain the same.

Continuous compounding is the limiting case of compound interest in which the compounding frequency approaches infinity. Its formula is derived by starting with the ordinary compound interest formula and applying a fundamental limit from calculus.

The final formula is:$$\boxed{{\displaystyle A=P{e}^{rt}}}$$