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Continuous Compound Interest Formula

The continuous compounding formula is used to determine the interest earned on an account that is constantly compounded, necessarily leading to an infinite amount of compounding periods.

The effect of compounding is earning interest on investment, or at times paying interest on debt, that is reinvested to earn additional monies that would not have been gained based on the principal balance alone. By earning interest on prior interest, one can earn at an exponential rate.

The continuous compounding formula takes this effect of compounding to the furthest limit. Instead of compounding interest on a monthly, quarterly, or annual basis, continuous compounding will efficiently reinvest gains perpetually.

\[\LARGE A=Pe^{rt}\]

Where,

A = Amount of money after a certain amount of time
P = Principle or the amount of money you start with
e = Napier’s number, which is approximately 2.7183
r = Interest rate and is always represented as a decimal
t = Amount of time in years

Solved Examples

Question 1: An amount of Rs. 2340.00 is deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.
Solution:

Use the continuous compound interest formula,
Given P =2340
r = $\frac{3.1}{100}$ = 0.031
t = 3

Use the continuous compound interest formula,
$A=Pe^{rt}$

Given,
$P=2340$
$r=3.1=\frac{3.1}{100}=0.031$
$t=3$

Here: e stands for the Napier’s number, which is approximately 2.7183.

However, one does not have to plug this value in the formula, as the calculator has a built-in key for e. Therefore,

$A=2340\;e^{0.031(3)}\approx 2568.06$

So, the balance after 3 years is approximately Rs. 2,568.06.

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