Intuition for continuous compound interest

Question

The formula for continuos compound interest is $$A = Pe^{(rt)}$$ Where $r$ is the interest rate per year. Interestingly, even though the money is compounded continuosly , it still depends on the annual interest rate rather than instantaneous interest rate. Is there any intuition to understand this?

Answer 1

For a principal $P$ and an annual rate of interest $r$ (compounded annually), the amount at the end of $t$ years is given by

$$A = P(1 + r)^t$$

Part 1

Before we move ahead, it's important to understand the meaning and significance of the phrase "compounded annually". The phrase implies that at the end of every year (because "annually"), we will take the amount at that point (principal at the start of that year + interest accrued during that year) and make it the principal for the next year.

If it said compounded semi-annually, it would imply that the end of every six months, we will take the amount at that point (principal at the start of that six-month period + interest accrued during that six-month perios) and make it the principal for the next six months.

Notice something? Frequency of compounding has nothing to do with the rate of interest. It just determines how often we will revise the principal (by adding interest to it) for the calculation of subsequent interests.

I hope this removes any doubts you had about why we are using the annual interest rate even when compounding continuously.

In our formula though (from above), $r$ will be replaced by $r/n$, where $n$ represents the number of times compounding in done every year. Why, you might ask? That's because the period for which we are calculating the interest is $1/n$ years.

As an example, if rate of interest is $10 \%$ and compounding is done semi-annually, the interest for each period would would only be $10 / 2 = 5$ percent because our period is only half a year.

So our formula becomes

$$A = P \left( 1 + \frac{r}{n} \right)^{nt}$$

Part 2

Higher the frequency of compounding, the faster your interest starts earning interest. That, in turn, would mean a higher amount at the end of a year or $t$ years.

But how high can we go? What is the maximum amount we can obtain by reducing the period of compounding only (leaving the interest rate unchanged)? Well, the best we can do (at least, theoretically) is compound an infinite number of times. So make $n$ approach $\infty$. That would give us an amount of

$$\begin{align*} A &= \lim_{n\to\infty} P \left( 1 + \frac{r}{n} \right)^{nt} \\[0.3cm] &= Pe^{rt} \end{align*}$$

Which, not surprisingly, is the formula you have in your post.

To conclude, the rate of interest and the frequency of compounding are two different things with nothing to do with one another. And the formula you mention is concerned only with the former.