One thing I don't see mentioned in TFA (I quickly skimmed it) is the relationship of the rule to continuous compounding. The article does say that it works better for "small R", after all - if interest rates where 100%, it should obviously take a year to double, not .72 years, right?
Well...no. Theoretically, if interest is 100% paid continuously throughout the year, you'll double your money in about .72 years. The issue is that we generally quote interest rates as annualized rates _after_ taking into account the compounding frequency (this is the difference between "APY" and "APR"), so the rule is usually applied to annualized rates, in which case it's correct to say that the error increases as the rates increase.
But suppose rates were actually quoted as the annual rate before compounding frequency is accounted for.
With monthly compounding, your return is
> (1 + r/12)^(12n)
where r is the annual rate and n is the number of years.
With weekly compounding, your return is
> (1 + r/52)^(52n)
Daily:
> (1 + r/365.25) ^ (365.25n)
More generally, if p is the number of interest periods in the year
> (1 + r/p) ^ pn
Taking the limit of p as it tends to infinity (ie "continuous compounding) it can be shown than the rate is
> e^rn
Some text books write this as "Pe^rt", where P is your starting principle and t is the number of years. Same same. Anyway...
To double your money, solve the following for n:
> e^rn = 2
> n = ln(2) / r
> n = ~.69 / r
Thus the rule of ln(2) is actually perfectly accurate if you're looking at an annual interest rate that is paid continuously, or close to it. Round ln(2) to 72 for simplicity and it's still pretty close. The big error bar comes from the lack of continuous compounding, which is what forces the hand-waving in the articles derivation ("for small R...").
P.S. Written off-the-cuff, it's been years since I looked at the math for this and I might be totally way off. This is not investment advice, this is hacker news and I'm positive some kind soul will correct my mistakes. After all, the best way to get correct information on the internet is to post something wrong on a forum of smart pedantics.
Yeah, you're right. I'm just talking about spherical chickens here. In real life, if your bank quotes you an APY they've already annualized it. I just think it's interesting to think about how in a perfect world, the rule is actually really good. Reality is just way less elegant than a classroom, unfortunately.
So I should have been clearer that I'm not criticizing the article, just offering another perspective on the how and why the rule is imperfect, which the article acknowledges as well.
The article is taking the first-order approximation of the doubling equation for discrete compounding interest, and as it turns out this value does not change no matter the compounding frequency.
For different compounding frequencies, the validity of the approximation is governed by how close r/p is to 0. As p goes to infinity, the approximation becomes arbitrarily good, converging to the continuous case.
I love the rule of 72, but my calculus class affectionately called it the rule of 69.pi (69.314). More accurate (you can round to 72 in your head anyway) and so much more fun to remember.
It's not necessarily more accurate to use 69.3. If you expand the Taylor series to two terms you get an even better formula, 69.3/x+0.34. Write it as (69.3+0.34x)/x and it shows how, as the interest rate grows, using a dividend in the 70–72 range is more precise than 69.3.
[That's another way to write the paragraph "A note on accuracy" from the article]
Here's what I don't understand about this: you won't do the long division in your head, so clearly, you'll have to memorise, for common interest rates, that 72/2 = 36, 72/3 = 24, 72/4 = 19, 72/5 = 15, and so on.
At that point, why wouldn't you just remember that "at 2 %, compounding doubles in 36 periods" and so on? Why the indirection through 72?
(Obviously, if you need a more accurate calculation or you don't remember the division, you'll probably use an electronic calculator, and then you might as well compute the compounding directly. The only exception I can think of is if you don't have access to an electronic calculator but do have a slide rule at hand. This is very rare these days.)
Many people do actually do the division in their head, I certainly do and not just for the trivial divisors either. In fact, I would find it odd if any of my mathematically, scientifically, or financially inclined friends did not do it in their heads, at least for quick calculations, to the point where I actually find your position that people would just memorize it instead of calculating it kind of shocking. It is truly amazing how people can have such different life experiences.
Here's my one weird trick: work toward a PhD in mathematics, then get a postdoc, where you finally get assigned to teach courses like upper division undergraduate number theory. Thanks to this experience, I can tell you immediately what the factors of numbers like 6 and 15 are. 72 still takes some thinking, though.
I usually do it by approximation most of the time. I estimate and then I multiply to see how close I got.
Multiplication is easier. 720 / 13 for this example and 1.3% looks pretty easy this way IMO, you just need a few iterations. I immediately know the result of using 10, so I'm already down to 720/130, or 72/13. Which shows me I did something stupid because I could have started that way and not forgotten what "%" means in the first place... Anyway, using 5 as a very lazy first estimate gives me 5 * 13=65 - certainly manageable in most heads. Trying 6 gives me 78.
Looking at both results, which have about the same distance from the target value, next I'll try 5.5 and that's 5 * 13 + a half of 13, still easy, 71.5. Close enough. Multiply by ten to compensate for the initial simplification, I get 55. That's really close to the result (~53.7, using the rule of 72 it is ~55.4).
I try to segment such calculations into simpler ones, instead of doing one algorithm, and finding shortcuts. For the 5.5 I use the shortcut that it's half of ten which always is easy, and that I can split it into two easy parts. To me it's more fun that way, even if I could also do the algorithm. It's supposed to be fun and being lazy is my goal.
For division you can often find shortcuts as well. Dividing by two, and powers of two subsequently, is simple enough. Any number you can divide by makes the number you have left easier. If you start from two you get prime factorization. You just have to remember a few numbers to put them together again (i.e. multiply) afterwards. Or any number less than ten really. Here too I would try to decompose. To me merely applying the algorithm is just work and maybe counts as mental exercise, but the decomposition and finding of little shortcuts is more fun and an exercise too.
In terms of the 72 issue I guess it turns a general purpose skill (division) into a specific one (investment calculation). So to do it you brush up on division and unlock other skills, like I guess angles 360/6 and so on (which comes in useful at times)
> you won't do the long division in your head, so clearly, you'll have to memorise
I think this assumption is false; at least for me, doing these types of divisions in my head is doable, I think we actually had to be able to do these types of calculations mentally in elementary school.
Eg in your examples you say 72/2 = 36, so then 72/4 must be 18.
I think the number 72 was chosen specifically for this property, that it divides easily for common values of interest rate (1, 2, 3, 4, 6, 8, 9 all divide cleanly).
72 = 2*2*2*3*3 so if the divisor is a multiple of those factors, the result is too. This makes the division quite easy in many cases. It's also why I immediately realized your typo in 72/4=19.
It's already an approximation - you do not need to get precise with your division. Also - you should be able to do 72 divided by at least 2,3,4 (and estimate it for others), even by times tables/estimation and not long division.
Well...no. Theoretically, if interest is 100% paid continuously throughout the year, you'll double your money in about .72 years. The issue is that we generally quote interest rates as annualized rates _after_ taking into account the compounding frequency (this is the difference between "APY" and "APR"), so the rule is usually applied to annualized rates, in which case it's correct to say that the error increases as the rates increase.
But suppose rates were actually quoted as the annual rate before compounding frequency is accounted for.
With monthly compounding, your return is
> (1 + r/12)^(12n)
where r is the annual rate and n is the number of years.
With weekly compounding, your return is
> (1 + r/52)^(52n)
Daily:
> (1 + r/365.25) ^ (365.25n)
More generally, if p is the number of interest periods in the year
> (1 + r/p) ^ pn
Taking the limit of p as it tends to infinity (ie "continuous compounding) it can be shown than the rate is
> e^rn
Some text books write this as "Pe^rt", where P is your starting principle and t is the number of years. Same same. Anyway...
To double your money, solve the following for n:
> e^rn = 2
> n = ln(2) / r
> n = ~.69 / r
Thus the rule of ln(2) is actually perfectly accurate if you're looking at an annual interest rate that is paid continuously, or close to it. Round ln(2) to 72 for simplicity and it's still pretty close. The big error bar comes from the lack of continuous compounding, which is what forces the hand-waving in the articles derivation ("for small R...").
P.S. Written off-the-cuff, it's been years since I looked at the math for this and I might be totally way off. This is not investment advice, this is hacker news and I'm positive some kind soul will correct my mistakes. After all, the best way to get correct information on the internet is to post something wrong on a forum of smart pedantics.