# Continuous Compound Interest Calculator

Created by Tibor Pál, PhD candidate
Reviewed by Arturo Barrantes and Adena Benn
Based on research by
Garrett, S. An Introduction to the Mathematics of Finance: A Deterministic Approach 2nd Edition; 2013
Last updated: Oct 09, 2023

With the continuous compound interest calculator (or continuously compounded interest calculator), you can quickly compute the final balance of your investment or savings with interest compounded continuously.

Read further, and you will learn the followings:

• How to calculate compound interest continuously;
• The continuous compound interest formula;
• How to solve continuous compound interest problems;
• Continuous compound interest vs. compound interest; and
• How to convert annual interest rate to continuously compounded.

## What is continuous compound interest?

Before introducing the idea of continuous compound interest and demonstrating its power, let's get familiar with the fundamental concept of compound interest.

Interest compounding is a process when the lender calculates interest not only on the principal but also on the previously accumulated (compounded) interest. More specifically, when the lender calculates the interest, she adds it to the principal, which will be the base of interest calculation in the following period. The higher the frequency of the process, the faster your balance grows. With our compound interest calculator, you can easily compare different scenarios of frequencies.

Continuous compounding is the theoretical limit of the compounding frequency. In this case, the number of periods when compounding occurs is infinite, as compounding would happen in every possible moment. To see its mathematical background, read the section on Natural logarithm in our log calculator.

## The continuous compound interest formula

If the compound frequency is continuous, the formula for continuous compounding interest takes the following form, where $e$ stands for exponential constant:

${\rm FV} = {\rm PV} \times e^{r \times t},$

where:

• $\rm FV$ – Future value or the final balance;
• $\rm PV$ – Present value or the initial balance;
• $r$ – Annual interest rate; and
• $t$ – Number of years.

To compute the interest which was compounded continuously, you need to subtract simply the final balance from your initial balance.

## How to solve for r in continuous compound interest

To compute the continuous compound interest rate, you need to solve the previously introduced equitation for $r$. Since $r$ is the exponent, the calculation would be burdensome to conduct by hand. Instead, a so-called can be applied, a mathematical algorithm using an iteration procedure.

## How do you calculate interest compounded continuously?

To compute interest compounded continuously, you need to apply the following formula. Interest = (Initial balance × ert) - Initial balance, where e, r, and t stand for exponential constant, periodic interest rate, and the number of periods, respectively.

## FAQ

### What are the most important factors in continuous compound interest?

The most crucial factors that determine your final balance are the following:

• Initial balance;
• Interest rate; and
• Time length.

### Will my final balance compounded continuously be higher with daily compounding?

No. Your initial balance will rise at the fastest pace if we compound it continuously since it is the upper limit of compounding frequency.

### How do I calculate continuous compound interest?

Follow the steps below to compute the interest compounded continuously.

1. Take the exponential constant (approx. 2.718) and compute its value with the product of interest rate (r) and period (t) in its power (ert).

2. Compute the future value (FV) by multiplying the starting balance (present value - PV) by the value from the previous step (FV = PV × ert).

3. The continuously compounded interest is the difference between the future and present values (Interest = FV - PV).

Tibor Pál, PhD candidate
Initial balance