Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

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jimbomahoney
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Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 12, 2020 6:06 am

UPDATE 21st Feb - I'll continue to update all of my posts to ensure they're as accurate as possible. I'm continually experimenting and refining the model...

UPDATE 23rd Feb I've refined my model and believe it's potentially in its final state and will begin trading it for real. Details in this post.

[Disclaimer - I have nothing to do with BreakingTheMarket, nor do I know who runs it - I'm just an average Joe with a nerdy streak]

Hi all,

I've been fascinated by, and roughly using, a Permanent Portfolio for my investing for some time.

Having read "How I Found Freedom in an Unfree World" as well as all of Taleb's books, I'm a "believer" in our inability to predict the future / appreciate tail-risk etc.

However, as a data and graph nerd, I've also always been fascinated with things like market timing, asset weighting etc.

I've previously written a few backtesting scripts that attempted to time the market (summary, as many of you will probably know already, is that it can't be done).

However, this guy has come up with a beautiful, mathematical method to decide what weight to assign each of the classic assets in the Permanent Portfolio.

EDIT - I appreciate that one needs to go hunting on that website to piece together what he's actually doing, so for general perusal, I recommend looking for the posts that are not "Portfolio on mm-dd-YY" and instead examine the ones with specific titles. This is as good a place to start as any. EDIT - I've also added links to posts of interest in some of my other responses - see my posts below.

Despite being a novice R coder, I've managed to piece together all the information in his posts and create a script that appears to work for the assets available to me, and priced in my currency.

I thought I would post here, as you may all be interested in his methods (I've only seen a couple of posts here mention his site before). I'd also be intrigued to hear others' thoughts.

The risk, as far as I can see it, for me is three-fold:

1) I only have 9 years of data to backtest.
2) I'm using index funds, rather than ETFs, which adds additional trading lag. [Removed ETF vs. index funds reference, as it's not relevant to the discussion]
3) My code might be wrong - since his methods are quite complex, I've had to make a number of simplifications in terms of cross-asset correlations and asset weights.

Having said that, my script seems to work, as including asset correlation in the weights improves the CAGR and Sharpe when compared with the method that simply weights assets based on their volatility (SD) / annual return.

UPDATE 18th Feb 2020 - I added the ability to limit the rebalancing frequency and, fortunately, it doesn't seem to make much difference. Yes, if you were able to trade that day's results on the same day and rebalance daily, you would get the highest returns. However, testing with a rebalance frequency of 5 days -> 6 months doesn't seem to have a huge effect. It still beats an equal split of assets, as well as a 60/40 Stocks/Bonds or 40/20/20 Stocks/Gold/Bonds portfolio.

Lastly, this thread is not about my code, which is almost certainly full of errors / look-ahead bias / too little data, but about BTM's theory of Geometric Rebalancing.
Last edited by jimbomahoney on Sun Feb 23, 2020 10:46 am, edited 7 times in total.

dandinsac
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by dandinsac » Wed Feb 12, 2020 9:29 am

Thanks for sharing this site. I found it interesting as well.

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jimbomahoney
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 12, 2020 2:04 pm

[Disclaimer - I know, both from experience and from theory, that backtesting is to be taken with a HUGE pinch of salt]

Some examples of the output of my script, which is my novice attempt to recreate what BreakingTheMarket has described on his blog:

Asset Weights over time, taking into account correlation (dotted lines for data are without correlation applied for comparison; horizontal dotted are average):

Image

Average for the period is ~20% Gold (Blue line), 45% Stocks (Green Line), 35% Bonds (Red Line), 0% Cash (Purple).

Returns:

Image

The key lines to concentrate on are:

1) Green line = stocks
2) Black line = Permanent Portfolio, rebalanced to maintain 25% in each asset.
3) Medium blue line ("Optimal Correlated Weights").

Stats:

Image

Annual Returns, including comparison with a 60/20/20 split : (UPDATED)

Image

Summary

This method appears to give stockmarket-like returns, but with almost PP-like volatility, and could therefore be leveraged. Correlation definitely helps, but was a complete nightmare to code! Note that leverage here includes the cost of that leverage, which for me as a "naive" investor, means borrowing against my house via the mortgage or taking an unsecured loan. Both around ~2% currently. (2.9% for the loan).

Bonus graph

Just because I love graphs, as well as learning to code, this one is a test of ~15,000 random portfolios of various weights for each asset over the same period. I use it as a "sanity check". It's saying what would have been the best fixed ratio of assets to use over the period.

Image

Best Sharpe would have been 23% Bonds, 54% Stocks, 22% Gold, 1% Cash

"Optimal" (for me) would have been 15% Bonds, 58% Stocks, 27% Gold, 0% Cash.

Again, this thread is not about this code - I did this just to challenge myself to see if I could piece together what BTM is doing. It was a fun challenge, both given the nature of his somewhat cryptic / piecemeal snippets and given the nature of some of the mathematics involved. I'm neither an experienced coder, nor mathematician. There was a lot of this... :oops:
Last edited by jimbomahoney on Tue Feb 18, 2020 5:46 am, edited 5 times in total.

MotoTrojan
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by MotoTrojan » Wed Feb 12, 2020 3:08 pm

Non of these assets have a lack of past data. Why are you not using other proxies to gather decades of results throughout various market cycles and conditions?

I don't quite understand why you feel an ETF is less safe than a Mutual Fund. An ETN is one thing, but I can't understand what makes an ETF a concern.

rascott
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by rascott » Wed Feb 12, 2020 3:46 pm

You realize that an ETF is a fund, right?.... it's right there in the name.

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jimbomahoney
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 12, 2020 4:59 pm

rascott wrote:
Wed Feb 12, 2020 3:46 pm
You realize that an ETF is a fund, right?.... it's right there in the name.
You realize that this thread is about geometric rebalancing to minimise volataility, right? It's right there in the title.

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jimbomahoney
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 12, 2020 5:00 pm

MotoTrojan wrote:
Wed Feb 12, 2020 3:08 pm
Non of these assets have a lack of past data. Why are you not using other proxies to gather decades of results throughout various market cycles and conditions?
Because BTM has already done that for me.

I want to make sure I'm recreating the theory to apply to assets that I want to trade / relevant to the currency with which I'm working.
MotoTrojan wrote:
Wed Feb 12, 2020 3:08 pm
I don't quite understand why you feel an ETF is less safe than a Mutual Fund. An ETN is one thing, but I can't understand what makes an ETF a concern.
I just have this weird paranoia about ETF vs. OEICs (what I meant when I said "fund"). I guess it's based on thing like this.

Anyway, ETF vs. OEIC vs. Fund vs. whatever is not relevant to the discussion at hand, so I won't go off-topic again.
Last edited by jimbomahoney on Wed Feb 12, 2020 5:03 pm, edited 1 time in total.

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willthrill81
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by willthrill81 » Wed Feb 12, 2020 5:01 pm

jimbomahoney wrote:
Wed Feb 12, 2020 4:59 pm
rascott wrote:
Wed Feb 12, 2020 3:46 pm
You realize that an ETF is a fund, right?.... it's right there in the name.
You realize that this thread is about geometric rebalancing to minimise volataility, right? It's right there in the title.
You didn't address the question. Mutual funds and exchange traded funds are both funds.
“It's a dangerous business, Frodo, going out your door. You step onto the road, and if you don't keep your feet, there's no knowing where you might be swept off to.” J.R.R. Tolkien,The Lord of the Rings

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jimbomahoney
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 12, 2020 5:08 pm

Hmm, I suspect I should have known from other threads that this would get derailed into grammar / one-upmanship / holier than thou instead of staying on-topic.

I had hoped that the mathematics behind BTM's theory would appeal, combined with the PP, but I think the PP is frowned upon around here?

However, the method seems sound and is aimed at increasing CAGR and Sharpe, which I would have thought would have been the aim for most (all?) posters here.

Either way, could we pretty please stay on-topic with the discussion of BTM's theory, rather than my personal idiocy? :sharebeer
Last edited by jimbomahoney on Fri Feb 14, 2020 2:23 pm, edited 1 time in total.

rich126
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by rich126 » Wed Feb 12, 2020 5:28 pm

It seems like he is trading or rebalancing weekly. That seems excessive. In a taxable account it would be a mess. With all of the zero commissions around, at least that expense is gone.

How do the numbers look if you do something quarterly?

I started to go through his web site but haven't had time to go through it all. While I find the numbers interesting the frequent trading is definitely a concern. I also got the impression, maybe falsely, that he seemed to be rather full of himself. Rather that just document everything in a long post, he seemed to drag things out over months.

Unlike most here, I'm not a big indexer and think it is the only way to succeed, and honestly, going forward I think those returns will be poor although I hope I'm wrong.

Uncorrelated
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by Uncorrelated » Wed Feb 12, 2020 5:54 pm

The problem with this analysis is that it only looks at an extremely short time period, and it is considered impossible to accurately estimate the parameters involved. Your analysis boils down to a mean variance analysis, which is a very valid analysis (I use it often), but is very much garbage-in-garbage out and without an out-of-sample analysis, there is no way of knowing how much you overfitted.

To give some indication to how difficult it is to estimate the correlations between different assets, the paper Gold Returns by Robert J. Barro investigates the correlation and return of gold over a long time period. In the period 1975-2011, neither the returns nor the correlation of gold vs stocks or bonds is statistically significantly different from zero (at the 95% confidence threshold). There is also the argument that gold is a big case or recency bias, there are very good arguments for using silver and a bunch of other precious metals and commodities to diversify, but that hasn't really panned out lately. I would consider picking gold over a commodities index a sin comparable to picking amazon over total stock market.

So in summary, this analysis is only as useful as the underlying assumptions. I believe that there is no reasonable set of assumptions that result in the inclusion of gold in a portfolio, so.. there is that.

Also, there is no particular reason why optimizing for max sharpe ratio or max CAGR is useful. Your goal should be to maximize your personal utility (or risk aversion) over the probability distribution of outcomes. For very specific cases that rarely occur in practice, this coincidences with maximizing the CAGR. Your chart marks the optimal portfolio for your specific assumptions, but there is no way of knowing which one is optimal without knowing your individual risk tolerance. Depending on your risk tolerance the optimal portfolio might be further to the right or left.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Thu Feb 13, 2020 3:11 am

rich126 wrote:
Wed Feb 12, 2020 5:28 pm
It seems like he is trading or rebalancing weekly. That seems excessive. In a taxable account it would be a mess. With all of the zero commissions around, at least that expense is gone.

How do the numbers look if you do something quarterly?
He's posted about that here.

He's suggested, maybe on that page or elsewhere on the site, that due to the high frequency of trades, the portfolio should be ~300k.
Uncorrelated wrote:
Wed Feb 12, 2020 5:54 pm
The problem with this analysis is that it only looks at an extremely short time period, and it is considered impossible to accurately estimate the parameters involved. Your analysis boils down to a mean variance analysis, which is a very valid analysis (I use it often), but is very much garbage-in-garbage out and without an out-of-sample analysis, there is no way of knowing how much you overfitted.
Thanks, that's a valid concern too. Although the data I'm using is short, and I absolutely agree with needing as much data as possible (during my backtesting / momentum / market-timing days, I thought I had found a "magic" frequency that worked beautifully for one market, but when I tested it on another market over a long period of time, Buy n Hold would have been better! I'm now very sceptical of anyone suggesting that their X day / Y day moving average method works because they invariably run it over the past 20 years, which of course was two massive bull and bear markets).

BTM has tested over much longer time periods. He also addresses, somewhat, the GIGO issue.
Uncorrelated wrote:
Wed Feb 12, 2020 5:54 pm
Also, there is no particular reason why optimizing for max sharpe ratio or max CAGR is useful. Your goal should be to maximize your personal utility (or risk aversion) over the probability distribution of outcomes. For very specific cases that rarely occur in practice, this coincidences with maximizing the CAGR. Your chart marks the optimal portfolio for your specific assumptions, but there is no way of knowing which one is optimal without knowing your individual risk tolerance. Depending on your risk tolerance the optimal portfolio might be further to the right or left.
Agreed. Again, that's just my personal "bonus graph" for fun. The "Optimal" point is just what I consider optimal for me, plus it was fun trying to code the detection of that top left corner.

In a perfect scenario, we'd all shoot for infinite returns and infinite Sharpe Ratios though! :D

dandinsac
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by dandinsac » Thu Feb 13, 2020 9:39 am

Great work! After reading the BTM I’m impressed you were able to create this. The BTM weekly rebalance was much more dynamic than I expected.

For an individual investor, one would have to have a tax advantaged account with low trading fees, the willingness to spend time analyzing and rebalancing, and the discipline to stay with it. It’s probably not practical for more than just a few dedicated folks.
jimbomahoney wrote:
Wed Feb 12, 2020 2:04 pm
Just because I love graphs, as well as learning to code, this one is a test of ~15,000 random portfolios of various weights for each asset over the same
Best Sharpe would have been 23% Bonds, 54% Stocks, 22% Gold, 1% Cash

"Optimal" (for me) would have been 15% Bonds, 58% Stocks, 27% Gold, 0% Cash.
EDITED, this question was already answered above: For the equal split, did you periodically rebalance the portfolio as well, or was it static over the study period?

Your post got me thinking on how I manage my investments. I can’t really move the funds around weekly due to trading restrictions. But, I do have recurring investments that I could direct to specific areas. Right now, those investments are directed to the investment(s) that is “under” my target valuation(s). My approach seems lacking as I don’t ever reevaluate the targets themselves. The BTM blog shows that the asset allocation target vary a lot. And compared with the fixed 25% approach, BTM did much better.

Can the BTM methodology be applied to do a better job of directing recurring investments and doing monthly rebalancing? Or is simply rebalancing enough?
Last edited by dandinsac on Thu Feb 13, 2020 12:34 pm, edited 1 time in total.

snailderby
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by snailderby » Thu Feb 13, 2020 9:53 am

1. This is fascinating. Thanks for sharing.

2. Does he address the best way to leverage this portfolio, if someone wanted to do that?

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jimbomahoney
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Fri Feb 14, 2020 11:46 am

snailderby wrote:
Thu Feb 13, 2020 9:53 am
1. This is fascinating. Thanks for sharing.

2. Does he address the best way to leverage this portfolio, if someone wanted to do that?
Glad you like it too! I'm loving it! I finally have a mathematical way to assign asset weights!

Re: Leverage - yes, the technical details are in this post and this post.

It's also discussed here and here.

I managed to code (in R) an approximation of both the correlation and leverage methods. Again, this was seriously challenging but seriously fun!

I've added an annual returns bar chart to an earlier post showing the various assets, an equal split (rebalanced) and a 60/20/20 (Stocks/Bonds/Gold), again rebalanced as well as the BTM, BTM (correlation) and leveraged versions. Again, that's for the assets I'm choosing to use (Long UK Gilts, World Stock Market and Gold in GBP).

klaus14
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by klaus14 » Fri Feb 14, 2020 9:56 pm

can someone explain how their system works?
it looks like every friday, they look at weekly std and correlations of assets.
then they determine allocations to optimize what? expected (mean) geometric return?
so they use historical return expectations?
15% VFMF, 15% NTSX | 10% ISCF, 5% EFAV | 5% FNDE, 5% EMGF, 5% VEGBX, 5% LEMB | 15% EDV, 5% CD (5y), 5% I/EE Bonds | 10% GLDM

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jimbomahoney
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Sat Feb 15, 2020 8:29 am

klaus14 wrote:
Fri Feb 14, 2020 9:56 pm
can someone explain how their system works?
it looks like every friday, they look at weekly std and correlations of assets.
then they determine allocations to optimize what? expected (mean) geometric return?
so they use historical return expectations?
I've provided plenty of links in each of my posts to the more relevant entries on his blog.

e.g. Check the "GIGO" link from my previous post.

klaus14
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by klaus14 » Sat Feb 15, 2020 2:04 pm

jimbomahoney wrote:
Sat Feb 15, 2020 8:29 am
klaus14 wrote:
Fri Feb 14, 2020 9:56 pm
can someone explain how their system works?
it looks like every friday, they look at weekly std and correlations of assets.
then they determine allocations to optimize what? expected (mean) geometric return?
so they use historical return expectations?
I've provided plenty of links in each of my posts to the more relevant entries on his blog.

e.g. Check the "GIGO" link from my previous post.
i checked those links but still didn't get the algorithm. Have you managed to produce the numbers he posts?
Can you share your code?
it shouldn't be that complicated. inputs seem to be daily returns of 3 assets. and maybe return expectations.
15% VFMF, 15% NTSX | 10% ISCF, 5% EFAV | 5% FNDE, 5% EMGF, 5% VEGBX, 5% LEMB | 15% EDV, 5% CD (5y), 5% I/EE Bonds | 10% GLDM

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Ethelred
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by Ethelred » Sat Feb 15, 2020 2:47 pm

It's an interesting blog, but I am sceptical to some extent. I'm not sure all of his assumptions are reasonable, and I don't think economists do concentrate on arithmetic averages of growth rate instead of compound to the extent he claims. That said, the CAGR and maximum drawdown he claims from back testing are consistent and very impressive.

I too was unable to find the actual algorithm he used, but the inputs seem to be current returns, current volatility (standard deviation?) and current correlation between each asset. Not sure if current means daily or a shorter period than that.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by mjb » Sat Feb 15, 2020 4:24 pm

I read his blog. He has some good points about geometric mean and effects of rebalancing.

However, he has some math errors that really skew his results and he ignores anyone that points them out. Mainly his conversion from arithmetic to geometric.

Additionally, several of his assumptions aren't fully true and he is using a short time horizon.

All he is really doing is just a flavor of risk parity.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Sun Feb 16, 2020 8:15 am

klaus14 wrote:
Sat Feb 15, 2020 2:04 pm
i checked those links but still didn't get the algorithm. Have you managed to produce the numbers he posts?
Can you share your code?
it shouldn't be that complicated. inputs seem to be daily returns of 3 assets. and maybe return expectations.
No, he doesn't give the algorithm, just the explanation from which I attempted to code it.

Yes, I get very similar returns when I backtest using the same assets as him (SPY, TLT, GLD in USD). Sometimes I get better returns, sometimes worse.

I can't match his volatility numbers though - for example, over the period 2005 - 2019, he reports about a 7% annual SD, whereas I can only achieve about 9%. I think this is almost certainly because:

1) I don't know what period he's using to get trailing annual returns. I'm guessing 2-3 years. EDIT - Actually, I think he's using fixed values for arithmetic returns over the longest period possible. See my other post(s).
2) I'm not correcting for the correlations as well as him. I've had to cut corners to get it into code. UPDATE 21st Feb - Actually, my correlation correction is very good, but I've had to simplify the way it reduces the weight the lower-performing correlated asset and adds it back to the better-performing correlated asset.

No, I'm not going to share my code because if "it shouldn't be that complicated", then anyone could code it, right? :wink:

But seriously, I don't want to share the code because:

a) It's ugly because I'm a novice (I started teaching myself R about 6 months ago).
b) It's probably wrong, although it seems to approximate his results, which is reassuring. UPATE 21st Feb - Yup, it's still an ongoing project and I keep finding, mostly minor, issues.
c) Probably most importantly, it isn't in a structure suitable to provide to someone else and they simply be able to run it. It has some dependencies, hard-coded directories and personal details. I can't be bothered to simplify it so that others can use it.
Ethelred wrote:
Sat Feb 15, 2020 2:47 pm
I too was unable to find the actual algorithm he used, but the inputs seem to be current returns, current volatility (standard deviation?) and current correlation between each asset. Not sure if current means daily or a shorter period than that.
Yes, for inputs I'm using:

1) Trailing annual returns (2 - 3 years probably) 4.5 years (shorter leads to excessive whipsaw) to estimate "current" returns. Shorter leads to more extreme asset allocation swings. Longer leads to more lag in asset allocation.
2) "Current" volatility, which is between 20 and 50 days (I use 38 - 60, as I found for my dataset, that worked best, but I don't think it makes much difference beyond about 30 days).
3) "Current" correlation - I use the same period as volatility.
Last edited by jimbomahoney on Fri Feb 21, 2020 8:58 am, edited 1 time in total.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Tue Feb 18, 2020 5:39 am

[UPDATED 23rd Feb - After refining the model, I find the same as BTM - shorter rebalance period = better. Yes, I'm aware of the post over at OfDollarsandData, but BTM is correct that 2005 - 2019 is too short a period of time to see the effect. This post has been updated to reflect this.]

OK, since I've been concerned about two things with how I want to implement this:

1) Lag (At best, I can only trade on yesterday's data, and I'm using Index Funds, rather than ETFs, which adds another day of delay in buy/sell but has no trading fees).
2) Rebalance frequency - I don't really want to be trading every day, partly because of the lag and partly because, although I'm using Index Funds (OEIC), I obviously need to use a GLD ETF, which has a trading fee.

I've done extensive backtesting (since 1983 with Stocks, Cash, Gold and since 2007 with Stocks, Bonds and Gold) with the following limitations imposed:

1) 1 day lag to ensure I can only trade based on yesterday's data.
2) Various rebalancing frequencies - from 1 days to several years.

I'm relieved that the rebalancing frequency doesn't have a huge impact on returns. UPDATE 23rd Feb - Actually, after refining my model, rebalance frequency does affect returns in the way BTM describes.

Here is a rebalance test using three assets (Gold, Bonds, Cash) since 1983 using my version of BTM's model:

Image

Blue/grey are CAGR/Sharpe for a BTM model that includes correlations (to change asset weights accordingly) and Yellow/Orange are the same model, but ignoring correlations (simply assigning weights based on geometric returns).

It's clear that shorter rebalance = better and OfDollarsAndData is correct that there is also noise associated with it - i.e. it's just dumb luck as to whether a particular frequency works. The problem is that he's ONLY seeing the noise over a ~14 year period, whereas I'm seeing the signal and the noise when the period is long enough (~40 years in this case).

Compare the same test on a short dataset (4 years for example):

Image

Here is what OfDollarsandData is getting at, but you can see that BTM is also correct because there is the same clear, rapid drop as rebalance periods increase from 1 -> 5 days.

Due to the lack of data in this case, one could conclude (erroneously) that rebalance periods result in random returns. Hopefully these two datasets confirm that BTM is correct, and OfDollarsAndData is correct over a short timeline.
Last edited by jimbomahoney on Sun Feb 23, 2020 8:20 am, edited 4 times in total.

klaus14
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by klaus14 » Tue Feb 18, 2020 2:26 pm

jimbomahoney wrote:
Tue Feb 18, 2020 5:39 am
OK, since I've been concerned about two things with how I want to implement this:

1) Lag (At best, I can only trade on yesterday's data, and I'm using Index Funds, rather than ETFs, which adds another day of delay in buy/sell but has no trading fees).
2) Rebalance frequency - I don't really want to be trading every day, partly because of the lag and partly because, although I'm using Index Funds (OEIC), I obviously need to use a GLD ETF, which has a trading fee.

I've done extensive backtesting (since 1983 with Stocks, Cash, Gold and since 2007 with Stocks, Bonds and Gold) with the following limitations imposed:

1) 1 day lag to ensure I can only trade based on yesterday's data.
2) Various rebalancing frequencies - from 5 days to 120 days.

I'm relieved that the rebalancing frequency doesn't have a huge impact on returns. Yes, if you were able to trade that day's market daily, you would get higher returns.

However, since 2007, the BTM method with 1 day lag and 5 - 120 days rebalancing frequency returned an average of 9.3% CAGR, compared with 5.4% for an equal asset split or 7.1% with a 60/20/20 (Stocks / Bonds / Gold). There wasn't a clear pattern in rebalancing frequency, but 90 days was best with a 9.9% CAGR. I strongly suspect that the frequency is simply coinciding with the market itself over this relatively short time frame.

Since 1983, the BTM method (using only Stocks, Gold and Cash) with 1 day lag and 5 - 120 days rebalancing frequency returned an average of 9.6% CAGR, compared with 4.8% for an equal asset split or 7.3% with a 60/40 (Stocks / Gold). Again, there wasn't a clear pattern in rebalancing frequency, but 60 days was best with a 10% CAGR. Again, I suspect that the frequency is simply coinciding with the market itself.

Given that asset correlations tend to change very rapidly, there isn't such a large difference between simply weighting the assets based on (expected annual return - standard deviation) vs. the same with adjusting for correlation when rebalancing less frequently. However, if rebalancing daily, then correlation has a large effect.

Here's an example output using all four assets since 2007, lagged by 1 day and rebalanced every 90 days:

Image

Notice that leverage really hurts because of the 2008 stock crash and the 2012 gold crash. I strongly suspect that lower frequency rebalancing means that you cannot use leverage because the response time is too slow and the drawdowns hurt too much. However, my leverage calculations also take into account the cost of that leverage, which for me would mean borrowing by increasing my mortgage or taking an unsecured loan. (I can't be bothered to investigate leveraged ETFs).

Here's an example output using only three assets, but since 1983, lagged by 1 day and rebalanced every 60 days:

Image

It's reassuring that the BTM method beats every asset and both fixed asset weight strategies.
So weighting by "(expected annual return - standard deviation)" is an approximation of this system?
But the problem is, as of today std is actually higher than expected annual return :) i assume expected annual return is the long term geometric return (cagr). it can be approximated as forward earnings yield for stocks (plus inflation). for bonds it should be the current yield. and for gold it is 0% real.

How about weighting by (expected annual return/ std) (like sharpe)

Do you mind explaining the mechanics of "adjusting for correlations" ?

actually, i'll request for one last time, can you write down the pseudo code or the outline of how you start from your inputs and determine the allocation. for, let's say no leverage, and monthly rebalancing.

----

overall system looks like volatility targeting. but also factors in expected returns. that's why i like it. i want see what kind of international alloc it would produce.
Last edited by klaus14 on Tue Feb 18, 2020 2:37 pm, edited 1 time in total.
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by imak » Tue Feb 18, 2020 2:31 pm

jimbomahoney, Thanks for sharing your work on recreating this strategy.
Based on my limited understanding, it seems that the core aspect of this strategy which makes it work is 1) rebalancing frequency (geometric balancing), and 2) kelly criterion for leveraging (probably fractional kelly?).

Regarding rebalancing frequency, here is an interesting blog post which counters the argument that frequent rebalancing is better:
https://ofdollarsanddata.com/how-much-d ... cy-matter/

Regarding kelly criterion, I believe BTM's assumption on position sizing is based on volatility prediction, i.e. in short term, volatility trend will persist (low volatility predicts low volatility & high volatility predicts high volatility). However there are multiple periods in US market history where volatility spikes where sudden and unexpected (for example, 1987 crash, 2001 (9/11 terror attack), 2010 & 2015 flash crashes). In these cases, BTM's portfolio will be very sensitive to volatility based estimations of position sizing and leveraging (BTM max leverage is 3X). I suspect few days of such extreme volatility events may be enough to wipe out gains of this strategy compared to a total market index fund.
Asset Allocation: 30% US Mid-Cap Blend/Value, 30% US Small-Cap Blend/Value, 15% US REIT, 15% ex-US Small Cap, 10% Long-term treasuries; Accumulation Phase; Stay the course!

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Tue Feb 18, 2020 5:15 pm

klaus14 wrote:
Tue Feb 18, 2020 2:26 pm

So weighting by "(expected annual return - standard deviation)" is an approximation of this system?
Sorry, that was a lazy summary. I ought to have been more explicit in my terminology and kept it consistent with BTM.

I'll outline in more detail below, but really, I'm just restating what BTM has explained over several of his posts, which I have previously linked to. I don't believe my method / code is an "approximation" - I'm pretty sure I've recreated his results, but whether he would agree with the methods I've used to get there, I don't know. In addition, one of the reasons I've used the term "approximation" before is due to correlation, but I'll also outline that below.
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
But the problem is, as of today std is actually higher than expected annual return :) i assume expected annual return is the long term geometric return (cagr). it can be approximated as forward earnings yield for stocks (plus inflation). for bonds it should be the current yield. and for gold it is 0% real.
That's not how I've done my code and not how I intepret BTM.

"Expected annual return" is, in BTM parlance, "Arithmetic Return". I've tried to guess as to what method he's using to get this data, since I can't find it explicitly stated anywhere. From my testing, and from what I've pieced together from his site, he either uses a "fixed" value, for example the annual arithmetic return of the S&P over the longest time period he can find, or some sort of "rolling" window of a suitable length, which is what I'm using in my code. Since he states that 250 days is not sufficient to measure the expected return, which I can also confirm with my testing, I'm using ~a 5 year rolling window, since this seems to work best with the data I have. UPDATE 27th Feb - Actually, I'm now deeply suspicious about using a rolling window as this leads to large swings in returns and asset weights depending on the length of that window.

Any shorter, and the returns are very volatile, and any longer and they fall slowly. i.e. very roughly a inverted "U" shape curve, with noise at the beginning, where the window to estimate annual returns is short. As BTM says, the problem with his method, and presumably mine as a result, is that it can be slow to respond to rapid changes in annual returns. This is also alluded to in the post below your (above this) by imak. However, since there is a spread of assets (i.e. Permanent Portfolio-esque) , a crash in one asset doesn't significantly impact the entire thing, as can be seen in my charts (excluding leverage).

Gold is also done in the same way - a ~5 year rolling window of arithmetic annual return. UPDATE 27th Feb - Actually, I'm now deeply suspicious about using a rolling window as this leads to large swings in returns and asset weights depending on the length of that window.

Once you have this rolling window, simply subtract the recent annualised SD (since, as BTM states and I have also replicated, the window only needs to be ~30-50 days to get a good value for the daily SD) and voila - that's the Geometric return for that asset.
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
How about weighting by (expected annual return/ std) (like sharpe)
I've no idea, but I guess I could change my code to do a division rather than subtraction!
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
Do you mind explaining the mechanics of "adjusting for correlations" ?
BTM had very helpfully posted an image showing how the asset weighting should change depending on the correlation between the two of them. See the excel plot at the very end of this post - the comment about it being very non-linear.

From that, I was able to determine the equation by digitising it and pulling out a polynomial fit from Excel.

I have just done this today, which is an improvement on my previous method of simply using the equation Y=(1-X^7) and ignoring any correlations below zero as irrelevant to make adjustments to. This is what I used the term "approximation" in reference to my correlation correstions. However, it is now much closer to BTM's calculations, at least if that plot is accurate, but it seems to make logical sense.

Anyway, once you have the correlation, for example if it's high like 0.8, then (approximately) (1-0.8^7) ~ 21% of the correlated asset with the lower geometric return gets reassigned to the higher-performing.
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
actually, i'll request for one last time, can you write down the pseudo code or the outline of how you start from your inputs and determine the allocation. for, let's say no leverage, and monthly rebalancing.
OK, I'll see if I can work on something. No promises though!
imak wrote:
Tue Feb 18, 2020 2:31 pm
Based on my limited understanding, it seems that the core aspect of this strategy which makes it work is 1) rebalancing frequency (geometric balancing), and 2) kelly criterion for leveraging (probably fractional kelly?).

Regarding rebalancing frequency, here is an interesting blog post which counters the argument that frequent rebalancing is better:
https://ofdollarsanddata.com/how-much-d ... cy-matter/
Hah, that's funny because you've reminded me where I first came across BTM's blog - it was from Dollars & Data! Thanks!

Yes I saw that post of course, and I admit that my testing, as you can see, also shows that rebalancing periods don't make a massive difference, or at the very least, are random.

I don't believe that reblacning frequency is key to BTM's method, and I don't think he states that. My impression is the keys are:

1) Expected returns (Arithmetic).
2) Recent SD / Volatility.
3) Correlations.
4) The use of 4 distinct asset classes, which are generally uncorrelated.
imak wrote:
Tue Feb 18, 2020 2:31 pm
Regarding kelly criterion, I believe BTM's assumption on position sizing is based on volatility prediction, i.e. in short term, volatility trend will persist (low volatility predicts low volatility & high volatility predicts high volatility). However there are multiple periods in US market history where volatility spikes where sudden and unexpected (for example, 1987 crash, 2001 (9/11 terror attack), 2010 & 2015 flash crashes). In these cases, BTM's portfolio will be very sensitive to volatility based estimations of position sizing and leveraging (BTM max leverage is 3X). I suspect few days of such extreme volatility events may be enough to wipe out gains of this strategy compared to a total market index fund.
His backtesting doesn't show that, and don't forget that he's usually in a mix of four asset classes, so just like the Permanent Portfolio, a 50% (or more) fall in one asset is no big deal.

However, I can't get my leverage to match his, but I'm using more "expensive" leverage (a consumer loan) and subtracting the costs from the returns.

It's also still likely that, despite the constant tinkering and testing, my code isn't the same as his theory.
Last edited by jimbomahoney on Thu Feb 27, 2020 3:35 am, edited 1 time in total.

klaus14
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by klaus14 » Tue Feb 18, 2020 5:34 pm

jimbomahoney wrote:
Tue Feb 18, 2020 5:15 pm
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm

So weighting by "(expected annual return - standard deviation)" is an approximation of this system?
Sorry, that was a lazy summary. I ought to have been more explicit in my terminology and kept it consistent with BTM.

I'll outline in more detail below, but really, I'm just restating what BTM has explained over several of his posts, which I have previously linked to. I don't believe my method / code is an "approximation" - I'm pretty sure I've recreated his results, but whether he would agree with the methods I've used to get there, I don't know. In addition, one of the reasons I've used the term "approximation" before is due to correlation, but I'll also outline that below.
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
But the problem is, as of today std is actually higher than expected annual return :) i assume expected annual return is the long term geometric return (cagr). it can be approximated as forward earnings yield for stocks (plus inflation). for bonds it should be the current yield. and for gold it is 0% real.
That's not how I've done my code and not how I intepret BTM.

"Expected annual return" is, in BTM parlance, "Arithmetic Return". I've tried to guess as to what method he's using to get this data, since I can't find it explicitly stated anywhere. From my testing, and from what I've pieced together from his site, he either uses a "fixed" value, for example the annual arithmetic return of the S&P over the longest time period he can find, or some sort of "rolling" window of a suitable length, which is what I'm using in my code. Since he states that 250 days is not sufficient to measure the expected return, which I can also confirm with my testing, I'm using ~a 5 year rolling window, since this seems to work best with the data I have.

Any shorter, and the returns are very volatile, and any longer and they fall slowly. i.e. very roughly a inverted "U" shape curve, with noise at the beginning, where the window to estimate annual returns is short. As BTM says, the problem with his method, and presumably mine as a result, is that it can be slow to respond to rapid changes in annual returns. This is also alluded to in the post below your (above this) by imak. However, since there is a spread of assets (i.e. Permanent Portfolio-esque) , a crash in one asset doesn't significantly impact the entire thing, as can be seen in my charts (excluding leverage).

Gold is also done in the same way - a ~5 year rolling window of arithmetic annual return.

Once you have this rolling window, simply subtract the recent annualised SD (since, as BTM states and I have also replicated, the window only needs to be ~30-50 days to get a good value for the daily SD) and voila - that's the Geometric return for that asset.
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
How about weighting by (expected annual return/ std) (like sharpe)
I've no idea, but I guess I could change my code to do a division rather than subtraction!
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
Do you mind explaining the mechanics of "adjusting for correlations" ?
BTM had very helpfully posted an image showing how the asset weighting should change depending on the correlation between the two of them. See the excel plot at the very end of this post - the comment about it being very non-linear.

From that, I was able to determine the equation by digitising it and pulling out a polynomial fit from Excel.

I have just done this today, which is an improvement on my previous method of simply using the equation Y=(1-X^7) and ignoring any correlations below zero as irrelevant to make adjustments to. This is what I used the term "approximation" in reference to my correlation correstions. However, it is now much closer to BTM's calculations, at least if that plot is accurate, but it seems to make logical sense.

Anyway, once you have the correlation, for example if it's high like 0.8, then (approximately) (1-0.8^7) ~ 21% of the correlated asset with the lower geometric return gets reassigned to the higher-performing.
klaus14 wrote:
Tue Feb 18, 2020 2:26 pm
actually, i'll request for one last time, can you write down the pseudo code or the outline of how you start from your inputs and determine the allocation. for, let's say no leverage, and monthly rebalancing.
OK, I'll see if I can work on something. No promises though!
imak wrote:
Tue Feb 18, 2020 2:31 pm
Based on my limited understanding, it seems that the core aspect of this strategy which makes it work is 1) rebalancing frequency (geometric balancing), and 2) kelly criterion for leveraging (probably fractional kelly?).

Regarding rebalancing frequency, here is an interesting blog post which counters the argument that frequent rebalancing is better:
https://ofdollarsanddata.com/how-much-d ... cy-matter/
Hah, that's funny because you've reminded me where I first came across BTM's blog - it was from Dollars & Data! Thanks!

Yes I saw that post of course, and I admit that my testing, as you can see, also shows that rebalancing periods don't make a massive difference, or at the very least, are random.

I don't believe that reblacning frequency is key to BTM's method, and I don't think he states that. My impression is the keys are:

1) Expected returns (Arithmetic).
2) Recent SD / Volatility.
3) Correlations.
4) The use of 4 distinct asset classes, which are generally uncorrelated.
imak wrote:
Tue Feb 18, 2020 2:31 pm
Regarding kelly criterion, I believe BTM's assumption on position sizing is based on volatility prediction, i.e. in short term, volatility trend will persist (low volatility predicts low volatility & high volatility predicts high volatility). However there are multiple periods in US market history where volatility spikes where sudden and unexpected (for example, 1987 crash, 2001 (9/11 terror attack), 2010 & 2015 flash crashes). In these cases, BTM's portfolio will be very sensitive to volatility based estimations of position sizing and leveraging (BTM max leverage is 3X). I suspect few days of such extreme volatility events may be enough to wipe out gains of this strategy compared to a total market index fund.
His backtesting doesn't show that, and don't forget that he's usually in a mix of four asset classes, so just like the Permanent Portfolio, a 50% (or more) fall in one asset is no big deal.

However, I can't get my leverage to match his, but I'm using more "expensive" leverage (a consumer loan) and subtracting the costs from the returns.

It's also still likely that, despite the constant tinkering and testing, my code isn't the same as his theory.
thanks!

look at this for expected return:
https://breakingthemarket.com/the-ultim ... -strategy/
15% VFMF, 15% NTSX | 10% ISCF, 5% EFAV | 5% FNDE, 5% EMGF, 5% VEGBX, 5% LEMB | 15% EDV, 5% CD (5y), 5% I/EE Bonds | 10% GLDM

Hydromod
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by Hydromod » Tue Feb 18, 2020 7:51 pm

klaus14 wrote:
Tue Feb 18, 2020 5:34 pm
jimbomahoney wrote:
Tue Feb 18, 2020 5:15 pm
It's also still likely that, despite the constant tinkering and testing, my code isn't the same as his theory.
thanks!

look at this for expected return:
https://breakingthemarket.com/the-ultim ... -strategy/
I've been waiting for a better explanation myself since following the dollars and data link. I really dislike all of the graphical explanation without the math.

I tried to follow the linked example just now. I'm missing something here. The formula for optimal weight for two assets appears to be, if you follow the trail,

w1 = (m1 - m2 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

where m is arithmetic mean, s is standard deviation, and r is correlation.

m = g + s^2 / 2

where g is the geometric mean.

This formula matched his results for the original example here.

However, in the linked example there are two variances reported for both TLT and for SPY, and I don't get the reported TLT weight using either one. Even the calculated variances are off by a little, maybe from rounding.

I get something like 33% instead of 19% for the TLT weight.

However, I get 19% if the yield for TLT is set to zero.

Can anybody reproduce the calculation? I want to understand this approach.

As a further check, if I have this right, the 3x leveraged return using UPRO and TMF would be

w1 = ((m1 - m2)/3 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

This would be from multiplying both m and s by 3.

In this case, I calculate a TMF weight of 44%. Which is HEDGEFUNDIE's current weight!

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by PluckyDucky » Tue Feb 18, 2020 8:37 pm

To put it simply, it looks like he rebalances with the Kelly criterion and leverage. You don't need to actually use leverage due to the 3x stock and bond funds. So you allocate cash and gold then fit the rest using the leverage funds as needed

Correct?

ETA: how is he choosing weights?

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 8:23 am

PluckyDucky wrote:
Tue Feb 18, 2020 8:37 pm
To put it simply, it looks like he rebalances with the Kelly criterion and leverage. You don't need to actually use leverage due to the 3x stock and bond funds. So you allocate cash and gold then fit the rest using the leverage funds as needed

Correct?

ETA: how is he choosing weights?
Check the various links I've posted or his blog. I think part of the reason why his method hasn't been widely circulated is because he's using a blog method to discuss one aspect of the method at a time, so piecing the whole thing together is tricky.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 8:43 am

Hydromod wrote:
Tue Feb 18, 2020 7:51 pm

I've been waiting for a better explanation myself since following the dollars and data link. I really dislike all of the graphical explanation without the math.

[SNIP]

Can anybody reproduce the calculation? I want to understand this approach.
The key I'm also missing, but have been trying to discover through testing, is how he arrives at the arithmetic return for each asset.

Since the geometric return (and correlations) is the basis for the asset allocation, which in turn is derived from the arithmetic return minus the SD, the arithmetic return is the key.

He hasn't responded as to what he's using as the artithmetic return.

In this post, he describes in detail that ~20-50 days is sufficient to determine the SD, which I have also found (between about 38 and 60 seems to be optimal). However, he only hints that 250 days is insufficient to determine the arithmetic return.

In this post, it looks like he is using a fixed arithmetic return, which I have also been using at times. However, I don't like this because it's unchanging - for example, we've had ~30 years of bull market in bonds. Using even a 30 year arithmetic return for bonds does not reflect what they might do in future.

Hence, I prefer to have a moving window of arithmetic returns, which should therefore respond as prices fall. Knowing that 1 year is insufficient, I've tried various rolling windows to estimate the current arithmetic return. The problem, of course, is the larger this window gets, the shorter our dataset to test becomes. UPDATE 27th Feb - Actually, I'm now deeply suspicious about using a rolling window as this leads to large swings in returns and asset weights depending on the length of that window.

Using the data he's posted in the welcome page, as well as his reported 2019 returns, I ran tests with various length moving windows to calculate arithmetic returns.

My tests back his statement that 1 year is not enough - my returns are all over the place and do not match his.

Between 2 - 4 years, my results match his quite closely, at least in terms of annual returns. Longer than that and the returns start dropping, presumably because the window is too long to accomodate changes in the market.

The other difference between my method and his is that he seems to be using "live" data and executing trades the same day.

My method uses the previous day's closing price, so is subject to lag.

Back to the subject of rebalancing windows, I've tested various rebalance periods, which seem to agree partly with OfDollars&Data in that they are random, but partly with BTM in that there is a sweet spot, or that rebalance periods do affect returns.

Here's a backtest using the same assets as BTM, but only since 2007, as that's the only data I can get hold of:

Image

As you can tell, the rebalance period doesn't seem to strongly affect the returns, but (possibly) tends to favour "mid-long" periods.

Here's another backtest with more data (since 1983), but using only Gold, Stocks and Cash (my TLT dataset is limited):

Image

Again, there seems to be a trend that too short and too long = bad.

Since I'm more interested in the assets in my currency, and a global stock market index, I repeated the test using my assets. The BIG problem here is that my dataset is even more limited - only ~2 years. Therefore, changing the rebalance period will be strongly affected by whether that rebalance happens at a particular point in time (i.e. during a period when arithmetic returns and/or SD are changing rapidly).

Image

Here, the response is much "noisier", but seems to match the other results in that there is a "sweet spot" in the middle of the range.

[UPDATE 23rd Feb] Actually, after refining the model and using long enough datasets, there is a correlation between rebalance frequency and returns. See my previous post that has been updated.

If it makes any difference, the units for the rebalance are trading days, so 5 days = 1 full week. Apologies for the mixed axes in those plots! (Some are CAGR on the left, others on the right), plus the colours aren't matched!
Last edited by jimbomahoney on Thu Feb 27, 2020 3:36 am, edited 3 times in total.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 9:13 am

UPDATE 27th Feb - Actually, I'm now deeply suspicious about using a rolling window as this leads to large swings in returns and asset weights depending on the length of that window.

Here are some results from me "hunting" for the arithmetic return window.

This leads me to suspect that he's either using ~2 year moving window or simply a fixed arithmetic return based on the long-term returns of each asset.

Like I said, I do not like the idea of a fixed arithmetic return because, as we all know, past returns are no indication of future returns. I suppose he *could* be justified in using very long-term expected returns (e.g. 100 years) because that should include all sorts of markets. But again, what's to say that the next 100 years is like the previous?

Image

This table shows the current asset weights that my method suggests and compares them to the current BTM recommendation. His returns are ~9-10% for this period. Green highlights a good match, red a bad match.

So you can see that if I use a 2 year rolling window, my results match his closely.

I can only conclude that he's using a fixed value for the arithmetic return. I'll test with that and have a look...
Last edited by jimbomahoney on Thu Feb 27, 2020 3:36 am, edited 2 times in total.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 9:25 am

Think we have a "bingo".

I think BTM is using a fixed arithmetic return.

UPDATE 21st Feb:
klaus14 wrote:
Tue Feb 18, 2020 5:34 pm
look at this for expected return:
https://breakingthemarket.com/the-ultim ... -strategy/
Thanks, that is fuel to the theory that he's using some sort of fixed value.

After a quick Google, I find that gold has returned an average 4.11% (335% over 30 years), stocks have returned 10% and long-term treasuries are 5.5%.

Plugging them into my method yields the following recommendation:

14% Gold / 61% Stocks / 25% Bonds

As of now, BTM recommends:

12% Gold / 52% Stocks / 36% Bonds

I still hate the idea of using fixed arithmetic returns, but perhaps it's valid if the period is long enough. For example, I really don't like the idea of using 30 year returns for something like bonds or gold because they could change rapidly in the future (we've never seen negative bond yields or the QE that's currently occurring).

On the flip side, using a moving window will bias weights towards what has performed well over that specific period of time...

On the other flip side (is there another!?), using a suitable moving window gives me similar results to BTM with the benefit of the system being more responsive to changes in the market... for example, a 2-3 year window delivers similar returns and similar asset weights... (see previous post).
Last edited by jimbomahoney on Fri Feb 21, 2020 9:02 am, edited 2 times in total.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by Hydromod » Wed Feb 19, 2020 9:44 am

I had a post that self-deleted. This repeats, in case it shows up somehow.

I returned to the example, using the values in the plot rather than the table. It looks like the assumed bond yields are 1.1% and the assumed geometric equity returns are 5.7%, implying a P/E of 17.5. This gives the reported 19% weighting for TLT.

The other thing that struck me is that the formula only requires the difference between arithmetic means (or, with some manipulation, the difference between geometric means), not the actual values.

Is the difference between means something that might be more easily justified as being fairly consistent, perhaps as the risk premium for equities over bonds? Something like 4 to 5% might be reasonable; this range gave 16 to 23% weights to bonds in the example. I don't know if this range would be better applied to arithmetic or geometric.

May be worth backtesting.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 11:30 am

Well, after extensive testing on a range of dates / datasets, I'm going to embark on a Geometrically rebalanced portfolio using the following settings:

1) SD / Volatility window = 55 days.
2) Rolling arithmetic window = 5.5 years.
3) Rebalance period = 90 days.
4) Assets = GLD in local currency (GBP), Global Stock Market Index, Long UK Gilts, Cash.
5) Backtesting lag = 1 day, ongoing lag = 0 days so that I can implement faster. However, with a rebalance period of 90 days, it won't matter much.
6) No leverage, or possibly slightly small leverage using the remnants of a low-cost loan if I borrow to get the bathroom done or some such, or remortgage.

Testing suggests that this massively outperforms a Permanent Portfolio (too much cash) and is generally superior to a mixed portfolio of 60/20/20 (Stocks, Bonds, Gold).

Wish me luck!


UPDATE - Actually, I've (unsurprisingly) found some errors in my code and need to iron them out...
Last edited by jimbomahoney on Fri Feb 21, 2020 4:52 am, edited 1 time in total.

Dudley
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by Dudley » Wed Feb 19, 2020 12:07 pm

On the surface of it this seems interesting, but I wish someone were able to simply and clearly articulate how the asset allocation is chosen and the rationale for doing so ..

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 12:41 pm

Dudley wrote:
Wed Feb 19, 2020 12:07 pm
On the surface of it this seems interesting, but I wish someone were able to simply and clearly articulate how the asset allocation is chosen and the rationale for doing so ..
I'll take a stab...

1) Pick a mix of uncorrelated assets. Stocks, Bonds, Cash and Gold are a good selection.
2) Assign an expected return (arithmetic / simple / linear) for each. As per this thread, this could be a fixed value, such as 10%, 5.5%, 4.1% and 0.5%, or a suitably long "rolling window". UPDATE 27th Feb - Actually, I'm now deeply suspicious about using a rolling window as this leads to large swings in returns and asset weights depending on the length of that window.
3) Measure their recent ("current") volatility (standard deviation). The trailing 30-50 days is generally used. Convert that to an annual SD.
4) Subtract the SD (actually the SD^2/2) from the (airthmetic) annual return - this gives the geometric (log / real) return.
5) Weight the assets according to their geometric returns.
6) Include correlation. When the correlation between two assets increases to 1, reduce the weight of the asset with the lower geometric return.
7) Leverage - compare each asset to the risk-free return. Increase leverage if the geometric return is greater than the risk-free return.

The geometric return is used because volatility hurts returns. In an ideal world, an asset (or portfolio) would have an infinite return with zero volatility. As we all know, generally a higher-return asset has higher volatility. This also inhibits the use of leverage because drawdowns are going to be magnified.

Does that help?

I strongly recommend going through all of BTM's posts in chronological. There are not many if you ignore the "Portfolio Update" posts.
Last edited by jimbomahoney on Thu Feb 27, 2020 3:37 am, edited 3 times in total.

mjb
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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by mjb » Wed Feb 19, 2020 12:44 pm

Hydromod wrote:
Tue Feb 18, 2020 7:51 pm
klaus14 wrote:
Tue Feb 18, 2020 5:34 pm
jimbomahoney wrote:
Tue Feb 18, 2020 5:15 pm
It's also still likely that, despite the constant tinkering and testing, my code isn't the same as his theory.
thanks!

look at this for expected return:
https://breakingthemarket.com/the-ultim ... -strategy/
I've been waiting for a better explanation myself since following the dollars and data link. I really dislike all of the graphical explanation without the math.

I tried to follow the linked example just now. I'm missing something here. The formula for optimal weight for two assets appears to be, if you follow the trail,

w1 = (m1 - m2 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

where m is arithmetic mean, s is standard deviation, and r is correlation.

m = g + s^2 / 2

where g is the geometric mean.

This formula matched his results for the original example here.

However, in the linked example there are two variances reported for both TLT and for SPY, and I don't get the reported TLT weight using either one. Even the calculated variances are off by a little, maybe from rounding.

I get something like 33% instead of 19% for the TLT weight.

However, I get 19% if the yield for TLT is set to zero.

Can anybody reproduce the calculation? I want to understand this approach.

As a further check, if I have this right, the 3x leveraged return using UPRO and TMF would be

w1 = ((m1 - m2)/3 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

This would be from multiplying both m and s by 3.

In this case, I calculate a TMF weight of 44%. Which is HEDGEFUNDIE's current weight!
His calculation for arithmetic mean is wrong. It is not the generalized form and as a specific form it is incorrect as well. To prove it, just use Excel for arithmetic mean and geometric mean.

All he is doing is a more complex risk parity system. However, you will never match what he gets because a) he keeps his exact time ranges proprietary and b) some of his equations are wrong (and likely other ones are too)

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Wed Feb 19, 2020 1:16 pm

mjb wrote:
Wed Feb 19, 2020 12:44 pm
His calculation for arithmetic mean is wrong. It is not the generalized form and as a specific form it is incorrect as well. To prove it, just use Excel for arithmetic mean and geometric mean.

All he is doing is a more complex risk parity system. However, you will never match what he gets because a) he keeps his exact time ranges proprietary and b) some of his equations are wrong (and likely other ones are too)
I think it's the terminology that's leading to this issue and I don't believe it affects the outcome.

Take for example the following price series, each a year apart:

100
120
150
80
100
130
120

Yes, I totally get that the arithmetic mean return is 7.6% (take the average of each year's returns).

I also totally get the geometric mean return is 3.1% (take the total period return and raise to one over the total period etc.)

[EDITED FOR MATHEMATICS]
I also (think) I totally get the fact that he's saying "to get the geometric return, subtract the annual SD away from the arithmetic return" and that this does not equal the actual geometric return. (I calculate an annual SD of 29.8% from this data. To convert to "GTM Geometric Return", we square that and divide by two and subtract from the Arithmetic return = 7.6% - ((29.8%)^2)/2 = 3.17% - I suspect that this is your criticism, however see my next sentence for my interpretation of why this doesn't matter).
[/EDITED FOR MATHEMATICS]

However, since we are treating each asset the same way and weighting them relative to each other, I don't believe this affects the outcome, which is to decide asset weights.

Agree / disagree / my maths wrong?

I've also pretty much matched his asset allocation using fixed values for arithmetic returns, although I fully admit that this could be a fluke. I should backtest and see what I get each month that he's posted the allocations... Cool, that's another mini-project sorted! 8-)
Last edited by jimbomahoney on Fri Feb 21, 2020 9:04 am, edited 3 times in total.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by Uncorrelated » Wed Feb 19, 2020 3:37 pm

The academic approach is to use the arithmetic excess return, the return above the t-bill rate. This figure stays stable during periods of inflation/deflation.

I think the idea of calculating the arithmetic return based on anything less than the full sample is an extremely bad idea. There is a wide array of research that suggests that the monthly return of stocks has a correlation of zero with the return of the preceding month. Some academics claim it's impossible to create a better estimate than the average over the full sample.

What is the rationale for using gold, given that gold has a return and correlation with stocks/bonds that are statistically indistinguishable from zero. And that factors (for example, the value factor) have statistically significant positive return? How do you avoid survivorship bias with gold?

I don't think the graphs showing the rebalanced interval can be interpreted with any statistical confidence. The best resource I know about rebalancing is this page: https://www.aacalc.com/docs/when_to_rebalance. Don't forget transaction costs either, 0.5% per transaction seems to be a common figure.
jimbomahoney wrote:
Wed Feb 19, 2020 1:16 pm
Yes, I totally get that the arithmetic mean return is 7.6% (take the average of each year's returns).

I also totally get the geometric mean return is 3.1% (take the total period return and raise to one over the total period etc.)
His formula for calculating geometric from arithmetic assumes that the underlying distribution is normal and independently distributed. But if you are using a market timing algorithm, then chances are that you don't believe that returns are normal and independently distributed...
jimbomahoney wrote:
Wed Feb 19, 2020 11:30 am
6) No leverage, or possibly slightly small leverage using the remnants of a low-cost loan if I borrow to get the bathroom done or some such, or remortgage.
If your risk tolerance is so low you are unwilling to use 2-3x leverage, CAGR is unsuited as a performance metric.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by mjb » Wed Feb 19, 2020 5:22 pm

jimbomahoney wrote:
Wed Feb 19, 2020 1:16 pm
mjb wrote:
Wed Feb 19, 2020 12:44 pm
His calculation for arithmetic mean is wrong. It is not the generalized form and as a specific form it is incorrect as well. To prove it, just use Excel for arithmetic mean and geometric mean.

All he is doing is a more complex risk parity system. However, you will never match what he gets because a) he keeps his exact time ranges proprietary and b) some of his equations are wrong (and likely other ones are too)
I think it's the terminology that's leading to this issue and I don't believe it affects the outcome.

Take for example the following price series, each a year apart:

100
120
150
80
100
130
120

Yes, I totally get that the arithmetic mean return is 7.6% (take the average of each year's returns).

I also totally get the geometric mean return is 3.1% (take the total period return and raise to one over the total period etc.)

I also (think) I totally get the fact that he's saying "to get the geometric return, subtract the annual SD away from the arithmetic return" and that this does not equal the actual geometric return. (I calculate an annual SD of 5% for this data, which would mean a "geometric return" of 2.6%, or possibly / probably an annual SD of 8 or even 12%. I'm cool with converting daily SD to annual, but not for converting from a sequence of annuals into an average annual, either way, see my next sentence).

However, since we are treating each asset the same way and weighting them relative to each other, I don't believe this affects the outcome, which is to decide asset weights.

Agree / disagree / my maths wrong?

I've also pretty much matched his asset allocation using fixed values for arithmetic returns, although I fully admit that this could be a fluke. I should backtest and see what I get each month that he's posted the allocations... Cool, that's another mini-project sorted! 8-)
It is more likely a coincidence. We have a ridiculously small data set for investing relative.to statistics. His equations for geometric mean is likely a misapplication
(and wrong equation but closest real equation to what he puts forward) of a method for developing the lower bound true mean from a sample mean. This is likely why it still mostly works as it has created a risk parity profile on a lower bound arithmetic average of an asset.

The thing is, it isn't necessarily a bad approach as it likely ends up overweighting assets with above average sharpe ratios. At the same time, would you highly leverage and bet your portfolio on a method that doesn't match the philosophy of the method, has math errors, and is likely overexposed to recency bias.

Alternatively you could take the 80/20 equity/itt "Boglehead risky buy and hold" or the ~33/66 equity/itt "optimum risk parity portfolio", rebalance both in bands, and leverage via having a home mortgage or even using treasury futures if you are so risk tolerant.

Note that NTSX being 90% stock, 9.7% t-bills, and 60% return of a treasury portfolio minus the return of t-bills is basically option 1. Hedgefundie's method mark II is a tax and labor inefficient version of option 2 (i.e. tax advantages only).

Alternatively for option 2, you could do 50%-80% equities, 50%-20% municipal bonds, and 100% to 160% treasury futures, rebalance quarterly when rolling futures, and likely match either with less drawdown risk. Check it in portfoliovisualizer. Probably the most tax efficient and least crazy risky way to leverage.

Note that a Portfolio that is 33% equities, 66% short/int term muni's, and 66% treasury futures is a tax efficient and technically unlevered way of attaining a risk parity portfolio (or 80/20/20 for the buy and hold) as all futures are backed by bonds but take.advantage of tax free muni's and 60/40 ltcg/stcg of treasury returns for easier and tax efficient rebalancing.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by jimbomahoney » Thu Feb 20, 2020 3:25 am

Uncorrelated wrote:
Wed Feb 19, 2020 3:37 pm
I think the idea of calculating the arithmetic return based on anything less than the full sample is an extremely bad idea. There is a wide array of research that suggests that the monthly return of stocks has a correlation of zero with the return of the preceding month. Some academics claim it's impossible to create a better estimate than the average over the full sample.

What is the rationale for using gold, given that gold has a return and correlation with stocks/bonds that are statistically indistinguishable from zero. And that factors (for example, the value factor) have statistically significant positive return? How do you avoid survivorship bias with gold?

His formula for calculating geometric from arithmetic assumes that the underlying distribution is normal and independently distributed. But if you are using a market timing algorithm, then chances are that you don't believe that returns are normal and independently distributed...
Re: Full sample - As I've outlined elsewhere, this is what I believe BTM is doing (fixed arithmetic return over a long time period) with a larger dataset than mine. I've also explained why I don't like that idea. This leads me on to my reasoning for the allocation to gold...

Re: Gold - I'm a big believer in the world being in constant change and that the future is unknowable. I'm not a "prepper" or a preparing for an end-of-the-world event, but I am aware that in human history, civilisations rise and fall. We tend to view the last 100 years, for example, as a decent timeline for determining investment returns, but I humbly disagree. Interest rates have never been lower (I'm sure we've all seen the recent graphs that have been put around showing ~700 years of interest rate history) and the sort of financial tricks the central banks are trying to play has also never been done (e.g. negative yields on bonds). In addition, there are multiple countries that have financially collapsed, both recently and throughout history, with only those owning physical assets (not cash, but I'm unsure of how access to brokers / bonds / stocks handles economic collapse - I would suspect that redemptions would be severely limited or outright banned in an effort to stabilise things) able to pull something from the wreckage. I've also seen banks close to home (e.g. Europe) shut their doors when the going gets tough.

Re: Normal distribution - that is a very good point and not something I'd considered, which is foolish of me given my readings of some of Taleb's work! Thanks for that, I will see if I can have a think about how to handle it...
mjb wrote:
Wed Feb 19, 2020 5:22 pm
It is more likely a coincidence. We have a ridiculously small data set for investing relative.to statistics. His equations for geometric mean is likely a misapplication
(and wrong equation but closest real equation to what he puts forward) of a method for developing the lower bound true mean from a sample mean. This is likely why it still mostly works as it has created a risk parity profile on a lower bound arithmetic average of an asset.

The thing is, it isn't necessarily a bad approach as it likely ends up overweighting assets with above average sharpe ratios. At the same time, would you highly leverage and bet your portfolio on a method that doesn't match the philosophy of the method, has math errors, and is likely overexposed to recency bias.
All good points.

I'm glad that you somewhat agree that this method is not deeply flawed, even if some of the maths may be incorrect.

I agree with the leverage. I don't plan on using much if any.

I've already put my own "spin" on his method, based on my beliefs about the world (using a rolling arithmetic return window, as opposed to a fixed value for a longer period).

We can only ever make decisions based on what we believe is correct!

I'm enjoying the discussions and am grateful for people's input.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by imak » Thu Feb 20, 2020 12:27 pm

I wonder if it is possible to approximate this strategy, without leverage, simply by applying Max-Sharpe rolling optimization to permanent portfolio.
Attempting this in Portfolio Visualizer -> Rolling Optimization feature, we see below risk/return profile:
Time period: 1994-2020
Rolling look-back window: 1 year
Rebalance frequency: Monthly
CAGR: 8.91%
Stddev: 8.98%

Link to results:
https://www.portfoliovisualizer.com/rol ... tion4_1=25

Note that I am selecting VBMFX as short-intermediate bond fund in place of CASH - this might not be accurately representing permanent portfolio. Also note that I selected benchmark as 60/40 balanced fund which also shows similar performance in terms of sharpe ratio.
Asset Allocation: 30% US Mid-Cap Blend/Value, 30% US Small-Cap Blend/Value, 15% US REIT, 15% ex-US Small Cap, 10% Long-term treasuries; Accumulation Phase; Stay the course!

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:27 pm

I’d like to thank Jim for starting this thread. I’m impressed by his knowledge of the system. He’s responded quite well to everyone’s questions.

What he’s doing here is exactly what I was hoping would happen. I make no claim I’ve got this 100% figured out. I’m certain the strategy is very improvable from where I have it today.

I’m trying to provide the framework for this way of investing, and let others and create their own strategies around that framework. My inputs, while good, are not perfect. The more people experimenting with these ideas, the faster the whole system will improve, and investing knowledge in the world as a whole.

Thank you to everyone for contemplating these ideas.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:30 pm

dandinsac wrote:
Thu Feb 13, 2020 9:39 am
Great work! After reading the BTM I’m impressed you were able to create this. The BTM weekly rebalance was much more dynamic than I expected.

For an individual investor, one would have to have a tax advantaged account with low trading fees, the willingness to spend time analyzing and rebalancing, and the discipline to stay with it. It’s probably not practical for more than just a few dedicated folks.
jimbomahoney wrote:
Wed Feb 12, 2020 2:04 pm
Just because I love graphs, as well as learning to code, this one is a test of ~15,000 random portfolios of various weights for each asset over the same
Best Sharpe would have been 23% Bonds, 54% Stocks, 22% Gold, 1% Cash

"Optimal" (for me) would have been 15% Bonds, 58% Stocks, 27% Gold, 0% Cash.
EDITED, this question was already answered above: For the equal split, did you periodically rebalance the portfolio as well, or was it static over the study period?

Your post got me thinking on how I manage my investments. I can’t really move the funds around weekly due to trading restrictions. But, I do have recurring investments that I could direct to specific areas. Right now, those investments are directed to the investment(s) that is “under” my target valuation(s). My approach seems lacking as I don’t ever reevaluate the targets themselves. The BTM blog shows that the asset allocation target vary a lot. And compared with the fixed 25% approach, BTM did much better.

Can the BTM methodology be applied to do a better job of directing recurring investments and doing monthly rebalancing? Or is simply rebalancing enough?
Yes, this works best in a tax advantaged account. But works well in a taxable account too. Large chucks of the portfolio don't get touched unless things get extreme. It does work monthly, although the drawdowns are higher. See this for a very simple version rebalanced monthly.

https://breakingthemarket.com/the-ultim ... -strategy/

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:31 pm

mjb wrote:
Sat Feb 15, 2020 4:24 pm
I read his blog. He has some good points about geometric mean and effects of rebalancing.

However, he has some math errors that really skew his results and he ignores anyone that points them out. Mainly his conversion from arithmetic to geometric.

Additionally, several of his assumptions aren't fully true and he is using a short time horizon.

All he is really doing is just a flavor of risk parity.

Where did I ignore someone pointing out a perceived error in the math?

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:32 pm

jimbomahoney wrote:
Tue Feb 18, 2020 5:39 am
OK, since I've been concerned about two things with how I want to implement this:

1) Lag (At best, I can only trade on yesterday's data, and I'm using Index Funds, rather than ETFs, which adds another day of delay in buy/sell but has no trading fees).
2) Rebalance frequency - I don't really want to be trading every day, partly because of the lag and partly because, although I'm using Index Funds (OEIC), I obviously need to use a GLD ETF, which has a trading fee.

I've done extensive backtesting (since 1983 with Stocks, Cash, Gold and since 2007 with Stocks, Bonds and Gold) with the following limitations imposed:

1) 1 day lag to ensure I can only trade based on yesterday's data.
2) Various rebalancing frequencies - from 5 days to 120 days.

I'm relieved that the rebalancing frequency doesn't have a huge impact on returns. Yes, if you were able to trade that day's market daily, you would get higher returns.

However, since 2007, the BTM method with 1 day lag and 5 - 120 days rebalancing frequency returned an average of 9.3% CAGR, compared with 5.4% for an equal asset split or 7.1% with a 60/20/20 (Stocks / Bonds / Gold). There wasn't a clear pattern in rebalancing frequency, but 90 days was best with a 9.9% CAGR. I strongly suspect that the frequency is simply coinciding with the market itself over this relatively short time frame.

Since 1983, the BTM method (using only Stocks, Gold and Cash) with 1 day lag and 5 - 120 days rebalancing frequency returned an average of 9.6% CAGR, compared with 4.8% for an equal asset split or 7.3% with a 60/40 (Stocks / Gold). Again, there wasn't a clear pattern in rebalancing frequency, but 60 days was best with a 10% CAGR. Again, I suspect that the frequency is simply coinciding with the market itself.

Given that asset correlations tend to change very rapidly, there isn't such a large difference between simply weighting the assets based on (expected annual return - standard deviation) vs. the same with adjusting for correlation when rebalancing less frequently. However, if rebalancing daily, then correlation has a large effect.

Here's an example output using all four assets since 2007, lagged by 1 day and rebalanced every 90 days:

Image

Notice that leverage really hurts because of the 2008 stock crash and the 2012 gold crash. I strongly suspect that lower frequency rebalancing means that you cannot use leverage because the response time is too slow and the drawdowns hurt too much. However, my leverage calculations also take into account the cost of that leverage, which for me would mean borrowing by increasing my mortgage or taking an unsecured loan. (I can't be bothered to investigate leveraged ETFs).

Here's an example output using only three assets, but since 1983, lagged by 1 day and rebalanced every 60 days:

Image

It's reassuring that the BTM method beats every asset and both fixed asset weight strategies.

You are correct that leverage is much safer over the shorter rebalancing frequencies

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:33 pm

Hydromod wrote:
Tue Feb 18, 2020 7:51 pm
klaus14 wrote:
Tue Feb 18, 2020 5:34 pm
jimbomahoney wrote:
Tue Feb 18, 2020 5:15 pm
It's also still likely that, despite the constant tinkering and testing, my code isn't the same as his theory.
thanks!

look at this for expected return:
https://breakingthemarket.com/the-ultim ... -strategy/
I've been waiting for a better explanation myself since following the dollars and data link. I really dislike all of the graphical explanation without the math.

I tried to follow the linked example just now. I'm missing something here. The formula for optimal weight for two assets appears to be, if you follow the trail,

w1 = (m1 - m2 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

where m is arithmetic mean, s is standard deviation, and r is correlation.

m = g + s^2 / 2

where g is the geometric mean.

This formula matched his results for the original example here.

However, in the linked example there are two variances reported for both TLT and for SPY, and I don't get the reported TLT weight using either one. Even the calculated variances are off by a little, maybe from rounding.

I get something like 33% instead of 19% for the TLT weight.

However, I get 19% if the yield for TLT is set to zero.

Can anybody reproduce the calculation? I want to understand this approach.

As a further check, if I have this right, the 3x leveraged return using UPRO and TMF would be

w1 = ((m1 - m2)/3 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

This would be from multiplying both m and s by 3.

In this case, I calculate a TMF weight of 44%. Which is HEDGEFUNDIE's current weight!
The equation is in the footnotes for the max return of 2 assets.
https://breakingthemarket.com/optimal-p ... wo-assets/

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:34 pm

Hydromod wrote:
Wed Feb 19, 2020 9:44 am
I had a post that self-deleted. This repeats, in case it shows up somehow.

I returned to the example, using the values in the plot rather than the table. It looks like the assumed bond yields are 1.1% and the assumed geometric equity returns are 5.7%, implying a P/E of 17.5. This gives the reported 19% weighting for TLT.

The other thing that struck me is that the formula only requires the difference between arithmetic means (or, with some manipulation, the difference between geometric means), not the actual values.

Is the difference between means something that might be more easily justified as being fairly consistent, perhaps as the risk premium for equities over bonds? Something like 4 to 5% might be reasonable; this range gave 16 to 23% weights to bonds in the example. I don't know if this range would be better applied to arithmetic or geometric.

May be worth backtesting.
Good observation about the absolute value not mattering, only the relative values.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:37 pm

jimbomahoney wrote:
Wed Feb 19, 2020 12:41 pm
Dudley wrote:
Wed Feb 19, 2020 12:07 pm
On the surface of it this seems interesting, but I wish someone were able to simply and clearly articulate how the asset allocation is chosen and the rationale for doing so ..
I'll take a stab...

1) Pick a mix of uncorrelated assets. Stocks, Bonds, Cash and Gold are a good selection.
2) Assign an expected return (arithmetic / simple / linear) for each. As per this thread, this could be a fixed value, such as 10%, 5.5%, 4.1% and 0.5%, or a suitably long "rolling window".
3) Measure their recent ("current") volatility (standard deviation). The trailing 30-50 days is generally used. Convert that to an annual SD.
4) Subtract the SD (actually the SD^2/2) from the (airthmetic) annual return - this gives the geometric (log / real) return.
5) Weight the assets according to their geometric returns.
6) Include correlation. When the correlation between two assets increases to 1, reduce the weight of the asset with the lower geometric return.
7) Leverage - compare each asset to the risk-free return. Increase leverage if the geometric return is greater than the risk-free return.

The geometric return is used because volatility hurts returns. In an ideal world, an asset (or portfolio) would have an infinite return with zero volatility. As we all know, generally a higher-return asset has higher volatility. This also inhibits the use of leverage because drawdowns are going to be magnified.

Does that help?

I strongly recommend going through all of BTM's posts in chronological. There are not many if you ignore the "Portfolio Update" posts.
This is a fairly good summary of the overall concept.

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Re: Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

Post by breakingthemarkt » Fri Feb 21, 2020 5:42 pm

mjb wrote:
Wed Feb 19, 2020 12:44 pm
Hydromod wrote:
Tue Feb 18, 2020 7:51 pm
klaus14 wrote:
Tue Feb 18, 2020 5:34 pm
jimbomahoney wrote:
Tue Feb 18, 2020 5:15 pm
It's also still likely that, despite the constant tinkering and testing, my code isn't the same as his theory.
thanks!

look at this for expected return:
https://breakingthemarket.com/the-ultim ... -strategy/
I've been waiting for a better explanation myself since following the dollars and data link. I really dislike all of the graphical explanation without the math.

I tried to follow the linked example just now. I'm missing something here. The formula for optimal weight for two assets appears to be, if you follow the trail,

w1 = (m1 - m2 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

where m is arithmetic mean, s is standard deviation, and r is correlation.

m = g + s^2 / 2

where g is the geometric mean.

This formula matched his results for the original example here.

However, in the linked example there are two variances reported for both TLT and for SPY, and I don't get the reported TLT weight using either one. Even the calculated variances are off by a little, maybe from rounding.

I get something like 33% instead of 19% for the TLT weight.

However, I get 19% if the yield for TLT is set to zero.

Can anybody reproduce the calculation? I want to understand this approach.

As a further check, if I have this right, the 3x leveraged return using UPRO and TMF would be

w1 = ((m1 - m2)/3 + (s2 - r * s1) * s2) / (s1^2 + s2^2 - 2 * r * s1 * s2)

This would be from multiplying both m and s by 3.

In this case, I calculate a TMF weight of 44%. Which is HEDGEFUNDIE's current weight!
His calculation for arithmetic mean is wrong. It is not the generalized form and as a specific form it is incorrect as well. To prove it, just use Excel for arithmetic mean and geometric mean.

All he is doing is a more complex risk parity system. However, you will never match what he gets because a) he keeps his exact time ranges proprietary and b) some of his equations are wrong (and likely other ones are too)

Risk Parity typically has portfolio weights of:
Stocks: 16%-30%
Bonds: 55%-70%
https://mebfaber.com/2015/05/28/chapter ... ortfolios/

My portfolio averages about
Stocks: 50%
Bonds: 35%
https://breakingthemarket.com/geometric ... unlevered/

Kind of the exact opposite weighting wouldn’t you say? Have you ever seen a RP portfolio hold far more stocks than bonds?

And please, I'd love to know which equations you think are wrong, even the ones you only suspect are wrong. I might as well fix them so I don't confuse someone else.

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