- April 16, 2024
- Posted by: legaleseblogger
- Category: Related News
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## Understanding the Impact of Compounding Frequency on Investment Returns
Hello Everyone,
I’ve been revisiting a topic that has always confused me a little bit: the impact of compounding frequency on investment returns and especially its connection with the number *e*. Here’s the usual spiel: if you’re compounding 100% interest yearly, you’ll double up. But increase the compounding frequency, and you’re told the returns can exceed doubling, potentially approaching *e* times the investment when compounded continuously.
This always struck me as odd, mostly because I couldn’t reconcile this with the fact that in the world of stocks or commodities, where growth is indeed continuous, a 100% yearly return means your investment doubles—full stop. No *e* factor in sight.
It turns out my confusion came from an undisclosed simplification in the math of everybody who tried to explain me compounding. The real deal is that a monthly return isn’t just the yearly return sliced into twelve pieces—it’s slightly less. This is usually done because, for small enough numbers, the difference is small and can be ignored, and it makes the math simpler. But in this tiny difference between the real formula and the approximated one is what actually converges to e as compounding become continuous.
In short, any effect of compounding frequency is just an error caused by your formula being just an approximation. if it looks like the frequency of compounding matters, it might be a sign you’re leaning too hard on a simplified formula, and maybe should start using the correct one.
I’m curious—how many of you were confused in a similar way, so I wanted to take a quick poll to see how much this simplification is engraved in everybody’s basic understanding of compounding
View Poll
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I’m assuming your confusion is caused by a fact that some instruments offer monthly/quarterly compounding while still advertising yearly rate of return. You’re indeed correct in assessing that monthly return in this case is a bit less than 1/12th of a yearly number, because it adjusts for compounding, this observation however isn’t relevant to compounding that happens year on year, where such adjustment isn’t necessary.
The return of an investment is an empirical fact, it isn’t affected by how you express it. The return can be expressed as a rate of return and a compounding frequency. The rate of return depends on the compounding frequency but the return does not.
I learned how simple and compound interest work first as basic concepts in high school, and then in detail in university. So those are a second nature to me. The various Excel sheets I’ve created through the years all calculate interest/return properly, e.g. they don’t simply multiply the monthly return of an asset by 12 to get its yearly return. Most people, however, would simply multiply by 12, because they don’t know any better, just like I probably wouldn’t know some random physics/philosophy/history/literature/geography/chemistry/whatever fact. It’s just niche knowledge.
Your analysis is largely correct but misses the fact that the simplified formulas ([and various complex variations thereof](https://en.wikipedia.org/wiki/Day_count_convention)) are still being used in most cases involving actual interest being paid/received (bank accounts, bonds coupons, etc). The frequency effect (but not the compounding effect, obviously) is indeed dependent on such formulas that do mathematically unsound stuff like this:
* quarterly interest = annual interest / 4
* monthly interest = annual interest / 12
* daily interest = annual interest / 360 or 365
instead of “properly” calculating interest as
* quarterly interest = (1 + annual interest)^(1/4) – 1
* monthly interest = (1 + annual interest)^(1/12) – 1
* daily interest = (1 + annual interest)^(1/365) – 1
* or, if you really hate simplifications: (1 + annual interest)^(1/(365+97/400)) – 1 for the average Gregorian calendar day
which, when applied 4, 12, or 365[.2425] times, give the exact same number as applying the annual rate once (i.e. no frequency effect).
Why that is so may be interesting history, no doubt connected to the fact that there was insufficient computing power available when these conventions were being established. But it’s still the way of doing things and, ironically, the simplified formulas lead to more complex computations when you start asking more questions beyond “how much interest do I owe/get” (and even that can get complicated in practice; see link above).
So you need to take it into account and do the corresponding weird math for interest payments, incl. increased return for the same quoted interest rate at higher compounding frequencies, to the point of getting *e* instead of 2.0 after an year at 100% interest compounding over infinitely small periods, because that is indeed what you’ll get from a bank (if you find one that offers infinitely small compounding periods).
Of course, people also get confused by this and take the frequency effect to be something natural that would hold for other things with compound growth, independent of this simplification (which is not the case).