Feb 18, 2023

So, I have been stuck trying to figure out how to quickly and easily answer this question using only mental math, any guidance to thinking would be greatly appreciated, "You take \$10 and invest it, 4 years from now, it's worth \$50. Without using pen, paper, or calculator, determine the average yearly growth rate of said investment."

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I just tried it and I got pretty close. But you do need to be comfortable with square roots.

Basically, you are after a CAGR to get from 10 to 50 in 4 years. If you know your CAGR formula you want to do 5^(1/4)-1. Now, that is equal to taking the square root of 5 twice: 5^(1/4)=(5^(1/2))^(1/2). Sqrt of 5 is ~2.1 to ~2.2 (you know it has to be a little more than 2, since 2 squared is 4). Then the sqrt of ~2.1-2.2 has to be a little more than ~1.4 but less than ~1.5. Since 1.5^2 = 2.25 and 1.4^2 = 1.96, but closer to ~1.5. So, I went with ~1.45-1.47, giving me a CAGR of 45-47%. Actual CAGR was ~49.5%. It's not very hard, but it certainly is an annoying one.

You do need to know some multiplication tables (specifically your squared table). But yeah, outside of identifying that it is a double sq rt, you just need to know that 2^2 = 4, and that 1.4^2 = 1.96 and 1.5^2 = 2.25. Not too bad, but could be tricky in a live situation when you're under pressure.

You can also ballpark it from the get go, knowing it has to be growing fast and is likely doing so at an "easy rate". Start from 10 and just add half each year (you know it has to be way more than 10-20% a year, but it also cant be 100% a year, cause in 2 years you are already at 40). So, 10 > 15 > 22.5 > 33.75 > ~50. So yeah, CAGR is around 50%.

EDIT: I feel I should caveat all this with the fact that (a) I am comfortable doing this kind of maths in general because that's just how I naturally count and (b) this was done while casually browsing online without any pressure from an interviewer. Live would have been a different experience, of course.

Appreciate the detailed breakdown! I was thinking the same thing about memorizing square roots. Just need to practice them I guess!

To be fair, I memorised them when I was young (up to 50^2), but then I forgot most of them, and now just remember some of the easier ones. It is handy in triangulating numbers like that.

But either my 2nd way or FinanceIsWacc's way are also much quicker. Basic logic is the same as most mental maths: find the closest thing you are familiar with and approximate using that. For me that was squares, but the 72-rule for doubling also works pretty damn well.

DrApeman's way is probably the best.

If I really needed to do the mental math in a split second without thinking I would do 75 / 4 = 18.75% but that's roughly the CAGR to double. Since your return is 5x I'd multiply that by ~2.5 which gets you to ~47%

Appreciate your input as well! This problem is definitely just one of those weird ones I need to take a second to pause and think through.

The average yearly growth rate of the investment can be found using the rule of 72. Since the investment doubled twice in 4 years (from \$10 to \$20, and then from \$20 to \$40, and then from \$40 to \$50), the growth rate is approximately 18% per year.

Here's how to use the rule of 72 to calculate this:

Divide 72 by the number of years it takes for the investment to double. In this case, the investment doubled twice in 4 years, so it takes 2 years for the investment to double. Therefore, 72 divided by 2 is 36. This means that the investment grows by approximately 36% every 2 years, or approximately 18% per year.

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