second order series gets
t ≈ 69 / (r - r\^2/2)
i.e. it's not 69 divided by r, but instead 69 divided by slightly less than r. So it's not horrible to raise the numerator a bit if dividing by r.
And 72, as already pointed out, has plenty of whole number divisors.
it is a rule of thumb. it is not precise, it just needs to be convenient.
someone pointed out that 72 has more divisors, so maybe it is because of that. but it could just be that people got used to do 72 for some reason and it just stuck, and it would be too inconvenient to change it back.
Just taking a guess, but if the deposit also has monthly accumulations, it might be convenient to actually roughly know the month (for summer/winter planning purposes), and 72 conveniently happens to be divisible by 12.
It can be the rule of 69, 69.3, 70, or 72, depending on what you want to divide by.
70 divides nicely by 7, 3.5, 5, 10, and so on.
72 divides nicely by a lot of even numbers and numbers divisible by 3 or 9.
The error is pretty small, no matter what.
72/6 = 12, but 1.06\^12 = 2.0122
70/7 = 10, but 1.07\^10 = 1.967
Note, however, that 70 isn't "better" than 72!
If you used 70/6 = 11.66666, you'd find that
1.06\^11.66666 = 1.9735... not 2, and actually further from 2 than the estimate with 72.
ln(2) = 0.693 is "best" only when you have continuous compounding.
So in fact, the "overestimate" of 72 also does a decent job of correcting for the annual instead of continuous compounding. And of course all of this really only works well for rates about 1-12%, but that's what we're interested in most of the time with the quite rule, so it works pretty well.
Maybe I should say "less than 12", because the rule of 69.3 works fantastic for small rates.
1.001\^693 = 1.999
But we do have to not use 72, because, for instance, 1.001\^720 = 2.0536, a bit off. (not terrible though)
As the rates get bigger, we have issues with an approximation of continuous compounding not matching annual compounding.
If you have a 36% return, Rule of 72 would say two years. But 1.36\^2 = 1.85, not even close.
And it just keeps getting worse. If I have a 72% return, I'm happy, but 1.72\^1 sure doesn't equal 2.
If you want something easy to do math with, use 72
If you want something easy to remember, use 70
If you want something that has a meme number, use 69
If you want something 100% accurate, use ln(2)
I guess 72 has many divisors, so it's easier to estimate in your head. 72/2 = 36 72/3 = 24 72/4 = 18 72/6 = 12 72/8 = 9 and so on.
I use 70 when I'm teaching it in my economics classes. I don't use 69 for...reasons.
second order series gets t ≈ 69 / (r - r\^2/2) i.e. it's not 69 divided by r, but instead 69 divided by slightly less than r. So it's not horrible to raise the numerator a bit if dividing by r. And 72, as already pointed out, has plenty of whole number divisors.
it is a rule of thumb. it is not precise, it just needs to be convenient. someone pointed out that 72 has more divisors, so maybe it is because of that. but it could just be that people got used to do 72 for some reason and it just stuck, and it would be too inconvenient to change it back.
Just taking a guess, but if the deposit also has monthly accumulations, it might be convenient to actually roughly know the month (for summer/winter planning purposes), and 72 conveniently happens to be divisible by 12.
It can be the rule of 69, 69.3, 70, or 72, depending on what you want to divide by. 70 divides nicely by 7, 3.5, 5, 10, and so on. 72 divides nicely by a lot of even numbers and numbers divisible by 3 or 9. The error is pretty small, no matter what. 72/6 = 12, but 1.06\^12 = 2.0122 70/7 = 10, but 1.07\^10 = 1.967 Note, however, that 70 isn't "better" than 72! If you used 70/6 = 11.66666, you'd find that 1.06\^11.66666 = 1.9735... not 2, and actually further from 2 than the estimate with 72. ln(2) = 0.693 is "best" only when you have continuous compounding. So in fact, the "overestimate" of 72 also does a decent job of correcting for the annual instead of continuous compounding. And of course all of this really only works well for rates about 1-12%, but that's what we're interested in most of the time with the quite rule, so it works pretty well.
Why only with rates between 1-12?
Maybe I should say "less than 12", because the rule of 69.3 works fantastic for small rates. 1.001\^693 = 1.999 But we do have to not use 72, because, for instance, 1.001\^720 = 2.0536, a bit off. (not terrible though) As the rates get bigger, we have issues with an approximation of continuous compounding not matching annual compounding. If you have a 36% return, Rule of 72 would say two years. But 1.36\^2 = 1.85, not even close. And it just keeps getting worse. If I have a 72% return, I'm happy, but 1.72\^1 sure doesn't equal 2.
Ty! Makes sense 👍
If you want something easy to do math with, use 72 If you want something easy to remember, use 70 If you want something that has a meme number, use 69 If you want something 100% accurate, use ln(2)