Q.Gaurav deposited ₹5000 in an account paying 3% interest compounded continuously for 5 years.
i. Find the total amount at the end of 5 years.
ii. How long will it take for the money to double?

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

Continuous compounding: $A = 10000e^{0.08 \times 3} = 10000e^{0.24} = ₹12,712.49$

Notice how daily and continuous are almost identical — after about 365 periods, you're already very close to the limit. The continuous result is the ceiling; no amount of further splitting can beat it.

The Deeper Why

Continuous compounding is not just a finance trick. It's the natural description of any process where the rate of change is proportional to the current amount — population growth, cooling of a hot object, charging of a capacitor, spread of a virus. In all these cases, the quantity follows $y = y_0 e^{kt}$.

The formula $A = Pe^{rt}$ is the solution to the differential equation:

which says "the instantaneous rate at which money grows is proportional to the amount present." That's the true meaning of continuous compounding — growth that is always working on the current total, every moment.