Recently, xkcd had a strip saying compound interest is not that great – $1000 with 2% interest for 10 years only grows to $1219.

But if we take the same $1000, and use a more optimistic 4% compounded continuously for 30 years, then we get $3320. That’s an advantage of $1120 over simple interest. What’s going on?

Normally continuous compounding is introduced by thinking about compounding yearly, then monthly, then daily, then taking the limit.

Let’s look at it another way. It’s simple interest, plus the simple interest on the interest, plus the interest on the interest on the interest, … and so on.

Now if you invest $1 and compound it continuously at interest rate r for time t, the simple interest is $rt.

The interest on the interest is rt*rt / 2 (we divide by 2 because we don’t get all of the first $rt at once).

And the interest on the interest on the interest turns out to be rt*rt*rt / (2 *3). Put it all together, and we get

1 + rt + (rt)^2/2! + (rt)^3/3! + … = exp(rt)

Now, for our 30 years at 4%, rt = 1.2

The simple interest is $1000 * rt = $1200

The interest on the interest is $1000 * (rt)^2/2! = $720

The interest on the interest on the interest is $288

The interest on the interest on the interest on the interest is $86

The interest on the interest on the interest on the interest on the interest is $21

The interest on the interest on the interest on the interest on the interest on the interest is $4.

And we’re still not done!

We can go one more level and get another $0.71. What does interest on the interest on the interest on the interest on the interest on the interest on the interest even mean? It means 71 cents.

That seems pretty amazing to me. If you don’t think so, I guess you don’t like money.

And it also explains more intuitively why compounding is so unimpressive for 10 years at 2%. In that case, rt = 0.2, and the successive interest terms fall to 0 very quickly.

But forget about the large amounts of money. We’re talking about either taking infinitesimal limits, or taking an infinite sum of interest on interest on interest on interest. That’s hippy stoner talk, but that’s how the modern financial system works. What could be more esoteric than infinity, or more mundane than a banker? And yet, there they are, side by side.

Einstein might or might not have said that compound interest was the most powerful force in the universe. But even if it’s not magical, it’s certainly mysterious and astonishing.

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