Q.Ms. Rajni deposited ₹10,000 in a bank that pays 4% interest compounded continuously.

Continuous Compound Interest

The Intuition: What Does "Continuous" Even Mean?

You already know compound interest. If you invest ₹100 at 10% per year compounded annually, after one year you have ₹110. If it's compounded semi-annually, the bank calculates 5% after six months, then another 5% on the new amount — so you get a little more than ₹110 because the interest from the first six months itself earns interest in the second six months.

Now push that idea. What if it's compounded monthly? Daily? Every second? Every microsecond?

As the compounding frequency increases, the amount after one year grows — but it does not grow without bound. It approaches a specific limit. That limit is what we call continuous compounding. It's the mathematical ideal where interest is calculated and added at every instant, not at discrete intervals.

The Precise Statement

Let's formalise this.

Suppose you invest a principal $P$ at an annual interest rate $r$ (written as a decimal, so 10% = 0.10) for $t$ years.

If interest is compounded $n$ times per year, the amount is:

Now let $n$ become very large — compound every day ($n=365$), every hour ($n=8760$), every second ($n=31,536,000$). What happens to $A$?

The limit as $n \to \infty$ is:

where $e \approx 2.71828$ is Euler's number.

Why $e$ Appears

The connection comes from a famous limit:

In our compound interest formula, rewrite it:

As $n \to \infty$, the inner bracket approaches $e$, giving $A = Pe^{rt}$.

A Concrete Example

You invest ₹10,000 at 8% per year for 3 years.

Annual compounding: $A = 10000(1.08)^3 = ₹12,597.12$

Monthly compounding: $A = 10000(1 + 0.08/12)^{36} = ₹12,702.37$

Daily compounding: $A = 10000(1 + 0.08/365)^{1095} = ₹12,712.32$ …