Golden Ratio (also in Bogleheads logo) Fibonacci
 spdoublebass
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Golden Ratio (also in Bogleheads logo) Fibonacci
I recently discovered the Golden Ratio. To be honest, I had never heard of it before. I had to research how it is found in classical music (mainly Bartok, but others insist the ratio appears in Beethoven and Mozart) for my doctorate.
Of course being a new Boglehead, I immediately thought to search the forums and how this relates to investing. There was one thread:
viewtopic.php?t=2245
In it you see that the divisions of the Boglehead logo are almost exactly in relation to the Golden Ratio.
Also, the first breakdown of 100 using the golden ratio is 61.8/38.2 (these numbers are rounded to 1 decimal point). This looks very much like the 60/40 AA often recommended.
I looked it up on investopedia:
http://www.investopedia.com/articles/te ... 033104.asp
To be honest, this got way over my head.
I have no point here, other than just a question out of curiosity...
Does anyone use this ratio for their investing?
I was curious if one could devise an AA out of it. The ratio lends itself to 4 or 5 divisions (which could be funds) which would remain simplistic.
Note: I do not use this for my investing.
Of course being a new Boglehead, I immediately thought to search the forums and how this relates to investing. There was one thread:
viewtopic.php?t=2245
In it you see that the divisions of the Boglehead logo are almost exactly in relation to the Golden Ratio.
Also, the first breakdown of 100 using the golden ratio is 61.8/38.2 (these numbers are rounded to 1 decimal point). This looks very much like the 60/40 AA often recommended.
I looked it up on investopedia:
http://www.investopedia.com/articles/te ... 033104.asp
To be honest, this got way over my head.
I have no point here, other than just a question out of curiosity...
Does anyone use this ratio for their investing?
I was curious if one could devise an AA out of it. The ratio lends itself to 4 or 5 divisions (which could be funds) which would remain simplistic.
Note: I do not use this for my investing.
I'm trying to think, but nothing happens

 Posts: 3040
 Joined: Thu Aug 09, 2012 10:54 am
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Why not Pi, e, or any other irrational number?
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
All AAs are really irrational despite our reasons for choosing them, no?
If it’s not long term it’s small talk
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
I think of fibonocci in terms of beauty.
Often plaids or stripes will use fibonacci to make a pleasing visual. I don't think of finances as using this conceptmat all.
Google. Fibonacci in nature.. pretty cool.
Often plaids or stripes will use fibonacci to make a pleasing visual. I don't think of finances as using this conceptmat all.
Google. Fibonacci in nature.. pretty cool.
 spdoublebass
 Posts: 463
 Joined: Thu Apr 27, 2017 10:04 pm
 Location: NY
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
off the top of my head, those irrational numbers aren't found as often in life. Our physical body is proportional to .618, even our own logo, sunflower seeds on the flower (all flower pedals actually), Parthenon, if you divide the number of female bees by male bees in a hive you get 1.618, in the Torah or Bible the Arc of the Covenant was described in detail, the dimensions follow the golden ratio.
I'm not going to argue that one should use this for their portfolio. I was only asking. However, when you ask why not another irrational number, I say because I don't see them elsewhere.
I'm trying to think, but nothing happens

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Re: Golden Ratio (also in Bogleheads logo) Fibonacci
I guess it depends if you consider math part of life or not. Just because it doesn't present itself visually in a flower doesn't mean its not there.spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life. Our physical body is proportional to .618, even our own logo, sunflower seeds on the flower (all flower pedals actually), Parthenon, if you divide the number of female bees by male bees in a hive you get 1.618, in the Torah or Bible the Arc of the Covenant was described in detail, the dimensions follow the golden ratio.
I'm not going to argue that one should use this for their portfolio. I was only asking. However, when you ask why not another irrational number, I say because I don't see them elsewhere.
 spdoublebass
 Posts: 463
 Joined: Thu Apr 27, 2017 10:04 pm
 Location: NY
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Of for sure. To be honest, I was just surprised that it was close to a 60/40 AA and that our logo had the ratio as well. Also, since the ratio is new to me, I was just curious if anyone used it. Ha!barnaclebob wrote: ↑Tue Sep 12, 2017 3:44 pmI guess it depends if you consider math part of life or not. Just because it doesn't present itself visually in a flower doesn't mean its not there.spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life. Our physical body is proportional to .618, even our own logo, sunflower seeds on the flower (all flower pedals actually), Parthenon, if you divide the number of female bees by male bees in a hive you get 1.618, in the Torah or Bible the Arc of the Covenant was described in detail, the dimensions follow the golden ratio.
I'm not going to argue that one should use this for their portfolio. I was only asking. However, when you ask why not another irrational number, I say because I don't see them elsewhere.
I'm trying to think, but nothing happens
 oldcomputerguy
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Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Pi is found in each and every circle in existence. How can that be construed as "not as often"?spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life.
It’s taken me a lot of years, but I’ve come around to this: If you’re dumb, surround yourself with smart people. And if you’re smart, surround yourself with smart people who disagree with you.
 spdoublebass
 Posts: 463
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 Location: NY
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Very true.oldcomputerguy wrote: ↑Tue Sep 12, 2017 3:57 pmPi is found in each and every circle in existence. How can that be construed as "not as often"?spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life.
Now that you mention it, there is even a thing called PiPhi
I'm trying to think, but nothing happens
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
This is mostly mythology. The golden ration doesn't feature in the human body, it isn't even close to true that 'all flower pedals' feature it, it isn't in the design of the Parthenon, it isn't in the ratio between males and females in bee hives.spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life. Our physical body is proportional to .618, even our own logo, sunflower seeds on the flower (all flower pedals actually), Parthenon, if you divide the number of female bees by male bees in a hive you get 1.618, in the Torah or Bible the Arc of the Covenant was described in detail, the dimensions follow the golden ratio.
 nisiprius
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Re: Golden Ratio (also in Bogleheads logo) Fibonacci
The idea of "Fibonacci retraces" and other Fibonacci ratios in investing is, in my opinion, totally nuts. There are actually reasons why Fibonacci numbers are sometimes (not universally and not precisely) found in the structures of plants (pineapples, sunflowers, leaf arrangements) and a "mathemusician" named Vi Hart has done a couple of wonderful YouTube videos about it:
Doodling in Math: Spirals, Fibonacci, and Being a Plant
Part 1: https://www.youtube.com/watch?v=ahXIMUkSXX0
Part 2: http://youtu.be/lOIP_Z_0Hs
Part 3: http://youtu.be/14NdQwKz9w
None of the reasons why Fibonacci numbers occur in plants apply to investing.
Theories as to why the Golden Ratio is aesthetically pleasing are not hugely convincing, and in real life it is pretty hard to be sure that the most aesthetically pleasing proportion is really φ = 1.618, and not some other number of about the same size such as the square root of two = 1.414. A rectangle whose sides are in the proportion φ has the property that if you remove a square from it, what's left is a rectangle with the proportion φ, but why should that be especially pleasing? The roottwo rectangle has the property that if you chop it in half, both of the small rectangles are also roottwo rectangles, and that seems intellectually pleasing... and of course roottwo rectangles are the basis of European paper sizes.
Doodling in Math: Spirals, Fibonacci, and Being a Plant
Part 1: https://www.youtube.com/watch?v=ahXIMUkSXX0
Part 2: http://youtu.be/lOIP_Z_0Hs
Part 3: http://youtu.be/14NdQwKz9w
None of the reasons why Fibonacci numbers occur in plants apply to investing.
Theories as to why the Golden Ratio is aesthetically pleasing are not hugely convincing, and in real life it is pretty hard to be sure that the most aesthetically pleasing proportion is really φ = 1.618, and not some other number of about the same size such as the square root of two = 1.414. A rectangle whose sides are in the proportion φ has the property that if you remove a square from it, what's left is a rectangle with the proportion φ, but why should that be especially pleasing? The roottwo rectangle has the property that if you chop it in half, both of the small rectangles are also roottwo rectangles, and that seems intellectually pleasing... and of course roottwo rectangles are the basis of European paper sizes.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
 DaftInvestor
 Posts: 4064
 Joined: Wed Feb 19, 2014 10:11 am
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
The reason it happens in nature regularly is best explained (in my opinion) by Fibonacci's original rabbit observation:
https://www.livescience.com/37470fibon ... uence.html
https://www.livescience.com/37470fibon ... uence.html
Last edited by DaftInvestor on Tue Sep 12, 2017 4:29 pm, edited 1 time in total.
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Looks like a double URL in the link, try this https://www.livescience.com/37470fibon ... uence.htmlDaftInvestor wrote: ↑Tue Sep 12, 2017 4:24 pmThe reason it happens in nature regularly is best explained (in my opinion) by Fibonacci's original rabbit observation:
https://www.livescience.com/37470fibon ... uence.html
Ah, the immature rabbits don't breed. I like it.
If it’s not long term it’s small talk
 DaftInvestor
 Posts: 4064
 Joined: Wed Feb 19, 2014 10:11 am
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Thanks  fixed link above (The Vkey must have stuck when I hit CTRLV).mrc wrote: ↑Tue Sep 12, 2017 4:28 pmLooks like a double URL in the link, try this https://www.livescience.com/37470fibon ... uence.htmlDaftInvestor wrote: ↑Tue Sep 12, 2017 4:24 pmThe reason it happens in nature regularly is best explained (in my opinion) by Fibonacci's original rabbit observation:
https://www.livescience.com/37470fibon ... uence.html
Ah, the immature rabbits don't breed. I like it.
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
e is all about naturally occurring continuous compounding growth phenomena, all around us including money matters It is in fact the most relevant of the 3 as far as financial matters are concerned! The famous rule of 72 for money doubling is related to e!!spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life.
 spdoublebass
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 Location: NY
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
You are right. I should have checked the sources of what I was reading. After a little time on google I found out most of it is wrong. Thanks for pointing it out.
As I said above, I was just curious. What sparked my interest was the 61.8/38.2 which reminded me of 60/40.
Nothing in my thesis or research depended on this information. (thank God)
It's funny how loosely people make their math, so it will fit this ratio. In music, Bartok did in fact use this ratio because he wrote about it. To be clear though, he really was still ballparking the figures.
Still odd how much it shows up. Also, for the Record I had no idea it was in the DaVinci Code. I have never read the book nor seen the movie.
I'm trying to think, but nothing happens
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
This is an important observation  people love the idea of elegant patterns so much they tend to see them everywhere whether real or imagined. It's important to keep that in mind when you read about all the magical investment systems that are discovered in past returns  I don't know why we are so allergic to randomness.spdoublebass wrote: ↑Tue Sep 12, 2017 5:53 pmIt's funny how loosely people make their math, so it will fit this ratio.
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
PI and e are actually named out of all the transcendental numbers precisely because they have common application in real life.spdoublebass wrote: ↑Tue Sep 12, 2017 3:34 pmoff the top of my head, those irrational numbers aren't found as often in life. Our physical body is proportional to .618, even our own logo, sunflower seeds on the flower (all flower pedals actually), Parthenon, if you divide the number of female bees by male bees in a hive you get 1.618, in the Torah or Bible the Arc of the Covenant was described in detail, the dimensions follow the golden ratio.
I'm not going to argue that one should use this for their portfolio. I was only asking. However, when you ask why not another irrational number, I say because I don't see them elsewhere.
Possibly one of the most amazing "coincidences" in math is Euler's identity e*(i*pi) + 1 = 0 It combines those two transcendentals along with the multiplicative and additive identities in the integers and that mysterious imaginary i that makes a complete field of solutions to polynomial equations in one variable. e is important in arising as part of the solution of the differential equation dy/dx=ay which arises naturally in such phenomena as radioactive decay and many other subjects in nature. Also the Euler equation can actually be expressed as e*(i*t) = cos(t) + i sin(t) which is a very useful identity in analyzing electrical circuits and electromagnetic waves among others, generally any system with oscillatory properties.
The number of transcendental numbers altogether is larger than the number of some other kinds of numbers including integers, rational numbers and algebraic numbers. Incidentally the golden ratio is an irrational but not transcendental number.
None of this has anything whatsover to do with investing. I find that prime numbers are a better approach to investing as in 53/47 allocations and so on. However, I give you credit for suggesting an approach that is more sensible than the appallingly stupid 1/n approaches such as 50/50, 60/40 (which is a variation on 1/10), and so on.
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Nothing special about Fibo & the Golden Ratio.
"Finding" it in nature is just backforcing data to support wishful thinking.
Even moreso the bunk in that investopedia article.
Same thing can be done with many other ratios.... and conspiracy theories.
(in my opinion)
"Finding" it in nature is just backforcing data to support wishful thinking.
Even moreso the bunk in that investopedia article.
Same thing can be done with many other ratios.... and conspiracy theories.
(in my opinion)
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
That the limit of the Fibonacci series ends up as the solution to a certain polynomial equation that originates in a simple geometric construction is indeed an amazing coincidence (if it is one  there are articles on the subject).BHUser27 wrote: ↑Tue Sep 12, 2017 6:33 pmNothing special about Fibo & the Golden Ratio.
"Finding" it in nature is just backforcing data to support wishful thinking.
Even moreso the bunk in that investopedia article.
Same thing can be done with many other ratios.... and conspiracy theories.
(in my opinion)
You are absolutely correct that "finding" this in nature is basically bunk.
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
I'm splitting hairs, but I don't like your use of the phrase "originates in".
I prefer to state it this way...
A simple geometric construction can be made from a number that is both the limit of an interesting mathematical series and the solution to a simple polynomial equation. This is just math.
The fallacy is the assumption that this number is somehow defined by or found in a special (magical, golden,) shape that occurs frequently in nature.
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
No problem. I am just referring to the possibility that one starts with this drawing: https://www.google.com/search?q=golden+ ... TcTMPGb2lM:
which then leads to a quadratic equation to be solved for a certain irrational number 1.618 . . .
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
In The Value of Debt in Building Wealth the author actually uses the Fibonacci sequence to lay out his suggested approach...
The "basic unit' is 3 months of income. From that point he then says your targets for Phase 2 are:
1..."basic unit" as a cash reserve in a checking account
1...in a savings account
2...in retirement savings
3...in home equity
8...in net worth
13...in mortgage
21...in total assets
(You can see he plays a bit fast & loose with the actual sequence.)
And then in another stage (Phase 3, maybe? I forget) he says the targets are
1...cash savings
1..."other" tangible assets (car, jewelry, etc)
2...home equity
3...longterm nonretirement savings (college education, buying an investment property, etc)
5...retirement savings
8...mortgage
13...net worth
21...total assets
I, uh, didn't really buy it at all but the author seemed to really be into it. He included an entire appendix just to go into it in more detail. (I don't really recommend the book.)
The "basic unit' is 3 months of income. From that point he then says your targets for Phase 2 are:
1..."basic unit" as a cash reserve in a checking account
1...in a savings account
2...in retirement savings
3...in home equity
8...in net worth
13...in mortgage
21...in total assets
(You can see he plays a bit fast & loose with the actual sequence.)
And then in another stage (Phase 3, maybe? I forget) he says the targets are
1...cash savings
1..."other" tangible assets (car, jewelry, etc)
2...home equity
3...longterm nonretirement savings (college education, buying an investment property, etc)
5...retirement savings
8...mortgage
13...net worth
21...total assets
I, uh, didn't really buy it at all but the author seemed to really be into it. He included an entire appendix just to go into it in more detail. (I don't really recommend the book.)
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Are you sure this is not just a parody of all the other dumb investing books out there?AlohaJoe wrote: ↑Wed Sep 13, 2017 9:53 amIn The Value of Debt in Building Wealth the author actually uses the Fibonacci sequence to lay out his suggested approach...
The "basic unit' is 3 months of income. From that point he then says your targets for Phase 2 are:
1..."basic unit" as a cash reserve in a checking account
1...in a savings account
2...in retirement savings
3...in home equity
8...in net worth
13...in mortgage
21...in total assets
(You can see he plays a bit fast & loose with the actual sequence.)
And then in another stage (Phase 3, maybe? I forget) he says the targets are
1...cash savings
1..."other" tangible assets (car, jewelry, etc)
2...home equity
3...longterm nonretirement savings (college education, buying an investment property, etc)
5...retirement savings
8...mortgage
13...net worth
21...total assets
I, uh, didn't really buy it at all but the author seemed to really be into it. He included an entire appendix just to go into it in more detail. (I don't really recommend the book.)
A different view is that in the end there may be no harm done as any of several arbitrary schemes can accidently not be a bad answer.
But, it looks to me that the categories net worth and total assets don't even make any sense unless he has some very strange definition of those two things.
 saltycaper
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Re: Golden Ratio (also in Bogleheads logo) Fibonacci
1/n does seem arbitrary, but do you really see comparative value in the prime approach, or were you just commenting on human psychology? You actually motivated me to consider properties of prime numbers that might help the situation. Then I realized you probably were joking and thus had proven your point, even though nobody besides me knew you made an example of me until I confessed herein.dbr wrote: ↑Tue Sep 12, 2017 6:16 pm
I find that prime numbers are a better approach to investing as in 53/47 allocations and so on. However, I give you credit for suggesting an approach that is more sensible than the appallingly stupid 1/n approaches such as 50/50, 60/40 (which is a variation on 1/10), and so on.
Quod vitae sectabor iter?
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
Well, it is basically a joke but really irony intended to broaden the perspective on things. Especially it is intended to downplay the tendency to see asset allocation as a magic bullet that is critical to investing and to stop people wasting too much time on figuring out exactly what number to use.saltycaper wrote: ↑Wed Sep 13, 2017 12:42 pm1/n does seem arbitrary, but do you really see comparative value in the prime approach, or were you just commenting on human psychology? You actually motivated me to consider properties of prime numbers that might help the situation. Then I realized you probably were joking and thus had proven your point, even though nobody besides me knew you made an example of me until I confessed herein.dbr wrote: ↑Tue Sep 12, 2017 6:16 pm
I find that prime numbers are a better approach to investing as in 53/47 allocations and so on. However, I give you credit for suggesting an approach that is more sensible than the appallingly stupid 1/n approaches such as 50/50, 60/40 (which is a variation on 1/10), and so on.
I was thinking once that maybe asset allocations should be expressed as one digit rather than percents in two digits. You could say my stocks are 3 or my stocks are 8, meaning 30% or 80%. That way going from 4 to 5 or 5 to 6 doesn't seem like a big deal and there is not this false precision of being able to talk about one part in a hundred in asset allocation. It also occurred to me that this silliness about primes might actually be helpful if it meant that the acceptable allocations for stocks were .3, .5, or .7 of the total. That means that asset allocation really just breaks down to not too much in stocks, a lot in stocks, or something judiciously in between. One is also reminded of Benjamin Graham's advice to keep stocks between 25% and 75%. And, if someone really wants to argue the point, then an extraordinary argument for an extraordinary allocation would allow an individual to opt for all stocks or no stocks.
 saltycaper
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 Location: The Tower
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
That sounds very practical. Perhaps we need to move Graham's lower boundary to .2 to keep all the primes.dbr wrote: ↑Wed Sep 13, 2017 2:09 pm
I was thinking once that maybe asset allocations should be expressed as one digit rather than percents in two digits. You could say my stocks are 3 or my stocks are 8, meaning 30% or 80%. That way going from 4 to 5 or 5 to 6 doesn't seem like a big deal and there is not this false precision of being able to talk about one part in a hundred in asset allocation. It also occurred to me that this silliness about primes might actually be helpful if it meant that the acceptable allocations for stocks were .3, .5, or .7 of the total. That means that asset allocation really just breaks down to not too much in stocks, a lot in stocks, or something judiciously in between. One is also reminded of Benjamin Graham's advice to keep stocks between 25% and 75%. And, if someone really wants to argue the point, then an extraordinary argument for an extraordinary allocation would allow an individual to opt for all stocks or no stocks.
The way I was going was more like taking 1/n to absurdity. I was trying to find two numbers that sum to 100 and have the distinct primes 2 and 3, so that every suballocation beneath stocks/bonds could be broken down into either 2 or 3 parts at least 2 times. For instance, 36 works, since it can be broken down into either (18, 18) or (12, 12, 12) and both 18 and 12 can be broken down into 2 or 3 parts. But then I'd be using 64, which does not have 3 as a prime factor. 72 also would work, but 28 does not. Then I thought, perhaps this is not possible, because 100 is not divisible by 3. (I only chose 2 and 3 because some of my suballocations are in two parts, others in 3. Three levels might seem excessive, but this includes holding the same or similar funds at different institutions.)
I might settle on 64/36 for overall allocation someday, since they are perfect squares, and that pleases me.
Quod vitae sectabor iter?
Re: Golden Ratio (also in Bogleheads logo) Fibonacci
saltycaper wrote: ↑Thu Sep 14, 2017 3:14 pmThat sounds very practical. Perhaps we need to move Graham's lower boundary to .2 to keep all the primes.dbr wrote: ↑Wed Sep 13, 2017 2:09 pm
I was thinking once that maybe asset allocations should be expressed as one digit rather than percents in two digits. You could say my stocks are 3 or my stocks are 8, meaning 30% or 80%. That way going from 4 to 5 or 5 to 6 doesn't seem like a big deal and there is not this false precision of being able to talk about one part in a hundred in asset allocation. It also occurred to me that this silliness about primes might actually be helpful if it meant that the acceptable allocations for stocks were .3, .5, or .7 of the total. That means that asset allocation really just breaks down to not too much in stocks, a lot in stocks, or something judiciously in between. One is also reminded of Benjamin Graham's advice to keep stocks between 25% and 75%. And, if someone really wants to argue the point, then an extraordinary argument for an extraordinary allocation would allow an individual to opt for all stocks or no stocks.
The way I was going was more like taking 1/n to absurdity. I was trying to find two numbers that sum to 100 and have the distinct primes 2 and 3, so that every suballocation beneath stocks/bonds could be broken down into either 2 or 3 parts at least 2 times. For instance, 36 works, since it can be broken down into either (18, 18) or (12, 12, 12) and both 18 and 12 can be broken down into 2 or 3 parts. But then I'd be using 64, which does not have 3 as a prime factor. 72 also would work, but 28 does not. Then I thought, perhaps this is not possible, because 100 is not divisible by 3. (I only chose 2 and 3 because some of my suballocations are in two parts, others in 3. Three levels might seem excessive, but this includes holding the same or similar funds at different institutions.)
I might settle on 64/36 for overall allocation someday, since they are perfect squares, and that pleases me.
Yeah, I like the idea of being able to continue dividing down. It is a little bit of a fractal concept, which being mentioned should make it appealing as an investment strategy.