Fascinating take on a rebalanced Permanent Portfolio - BreakingTheMarket.com

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

Post by Uncorrelated » Sun Feb 23, 2020 4:13 pm

jimbomahoney wrote:
Sun Feb 23, 2020 1:39 pm
Uncorrelated wrote:
Sun Feb 23, 2020 11:25 am
It appears that we disagree on the definition of market timing, and also whether it is possible. If you are varying your asset allocation as a result in economic conditions, that is market timing.

It's not that I don't believe you or BtM, just that the arguments are either nonexistent or poor. This makes it nearly impossible to evaluate the strategy.
That's a very fair point, and I do agree that choosing a window of a fixed period is equivalent to market timing.

However, even a "window" of the entire dataset equates to the same thing. I'm not happy using a 30 or 100 year "window" to form an expectation of future returns. I've no idea what future reaturns are going to be, so any "window" is, at best, a guide.

I'm just going to have to accept the risk either way. A "fixed" expectation will not respond to changes. A moving window will be "biased" to whatever the dataset has tended towards.

The reason I like a moving window of returns, and the reason I'm fairly confident about it is two-fold:

1) My returns are similar to BTM, who as far as I can tell, is using a fixed return value for ~70+ years of data. I'm using a ~3-4 year moving window and getting similar returns, albeit with more volatility.
2) The fact that there is a pattern, similar to the rebalance frequency, that tends to be noisy for short periods (and therefore unreliable) but clearly trails off at longer periods. Yes, the short periods tend towards chance and the particularities of that dataset / time period. But there is, I believe, a trend. Short = random, but higher likelihood of being "optimal". Long = less noisy, but higher likelihood of lower returns. See some of the plots in my long post.

This is my justifcation anyway. I totally appreciate that it could be chance that 2 - 8 years just happens to work best for this time period.
If you believe that market timing is impossible, then using a window equal to the entire dataset is the best possible approach. This is the most unbiased estimator possible to sample the true underlying mean of the return distribution.

I don't find your reasons to be convincing. If a limited window is better than an estimate based on the full dataset, then the only explanations I can come up with are chance and momentum.
Uncorrelated wrote:
Sun Feb 23, 2020 11:25 am
The "optimal" window for calculating the arithmetic return with your data is 2.5-8 years. Are you calculating this based on the nominal returns, real returns, or the excess returns? What is the theoretical basis for this window size? What are the objections against using a window spanning the entire data period? I fear that using a short window may (accidentally) result in exploitation of momentum effects, you don't seem to believe in momentum.
I'm taking the arithmetic mean daily return for the period and converting it to an annual return. i.e. this should be the arithmetic annual return. I guess this is the nominal return.
I expected that was the case based on your estimated window size. This is a pretty big error, the returns should be calculated in excess of the risk free rate. The are three reasons for this: a log utility agent only cares about the return and volatility(!) in excess of the risk free rate. It is consistent with the theory of an equity risk premium, and it would make the strategy independent of the currency inflation. Since the dollar has seen widely varying inflation in the last 50 years, this greatly impacts the results.

There is some discussion over what the correct risk-free rate is, but the most commonly used is either the 1 month or the 3-month t-bill.
Uncorrelated wrote:
Sun Feb 23, 2020 11:25 am
Regarding the SD and correlation window length, the longer the correlation window is, the better the results. Doesn't that imply that the best estimate for correlation is an estimate over the full sample? (i.e. correlation is not time-varying). The results when varying the SD look pretty random and statistically insignificant. If you can't find a theoretical explanation why an 23-day SD window is better than a 10-day or 40-day SD window, I think there is a large chance that this is a result of overfitting.
Again, there is a trend. The results from SD window length become more stable as they get longer, but there is, as BTM says, not much difference once it's sufficiently long (e.g. 20 - 50 days). See my analysis in the long post.
That doesn't fully answer my question. I don't see any real evidence that estimating the SD based on a 20-50 day window is better than estimating the SD based on the entire dataset. Maybe it is. But I don't see any evidence.
I'd also considered that and can show some results. I'd deliberately designed in various checks as the code executes so that I can see what it's doing.
Based on these images I am almost certain that virtually all of the returns are explained by the accidental momentum exposure present in the algorithm. Furthermore, the data shows substantial serial correlation which means that the results are extremely sensitive to overfitting. With a return estimation window of 5 years, there are only 3 independent data samples present in the images you posted.

That's one data sample each for the train, test and validation set (that was a machine learning joke, but it's not far from the truth).

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

Post by breakingthemarkt » Sun Feb 23, 2020 5:28 pm

[/quote]

He's using a formula for optimal weights, at least for two assets, that is taken from a paper that assumes normally distributed and correlated returns with different means and variances. Not independent and identically distributed. One needs to go to the original source to get the theory. I'm wondering what one would get assuming the Laplace distribution instead.

[/quote]

Nice call on the Laplace distribution. You don't see many people acknowledging the that distribution's application to investment behavior often.
Last edited by breakingthemarkt on Sun Feb 23, 2020 6:04 pm, edited 1 time in total.

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

Post by breakingthemarkt » Sun Feb 23, 2020 5:33 pm

jimbomahoney wrote:
Sun Feb 23, 2020 10:38 am
I've been working on refining my model of BTM's theory so that it suits my situation and my beliefs about the world.

I've taken what BTM has done and modified it thus:

1) Cannot trade in "real-time" - i.e. must wait until at least the next day. This is partly due to the availability of real-time data on the assets I wish to use and partly because I do not want to trade daily. Therefore my model adds one day of lag so that it only trades based on yesterday's close. In code, I'm added a "lag" variable, which will basically offset the dataframe by n days.

2) Cannot trade too frequently - this is similar to the previous constraint, but additionally, I do not want to be trading daily. In code, I've added a "rebalance frequency" variable, which will basically extend the daily weights found from #1 by an additional n days.

3) UK assets - Long UK Gilts, Gold in GBP, Global Stock Market. I don't believe in investing in a single stock market index, hence prefer a global tracker, which is of course even more diversified than the S&P. I'm using Long UK Gilts only because I cann't find a suitable global bond index to test data on.

4) Moving arithmetic return window. I believe this differs from BTM's theory, but perhaps he will correct me if I'm wrong. As previously stated, I believe he uses a "fixed" value for the arithmetic return based on a long history of returns. Because of my beliefs about the future, I don't want to use a "fixed" arithmetic return as basis on which to derive expected returns, no matter how long the data from which that return is based.

My model allows me to vary the following:
  • Arithmetic return window
  • SD window
  • Correlation window
  • Rebalance frequency
  • Risk-free rate
  • Leverage cost
  • Cap on gold (from 0 - 100%)
This has allowed me to do extensive testing on the effects of each, both on the data I will use to execute my trades (a short dataset), the same assets that BTM uses (TLT, GLD, S&P, Cash, but again, over a relatively short timeframe of ~13 years) and also over a large dataset using only GLD, S&P and Cash (almost 40 years).

I'll try and explain each and the results of my testing:

Arithmetic Return Window

As I've stated numerous times, I really don't like the idea of a fixed value for the arithmetic return. On the other hand, if the period of time is too short (as BTM has also stated), the estimate will basically be garbage. So, using the largest dataset I have (40 years of gold and S&P), I tested the effect of using a moving arithmetic return:

Image

The X axis is in years.

Just as BTM suggests, anything less than ~2 years is garbage. However, there is a pattern here (see the curve of best fit I've plotted), which I don't believe is "random".

Yes, the returns I'm getting jump around, but I believe that, as long as the window over which to calculate arithmetic returns is "sensible" (between about 2.5 and 8 years), it can be used.

The same pattern emerges using the assets I plan on using, however, because my dataset is so short, as I increase the window, I "crop" my dataset, so have fewer years on which to test.

I've also tested this to examine the annual returns I get vs. BTM, since he's posted his results a few times on his blog:

Image

Bearing in mind that those results are with the constraints of lagging the market by one day, I think that's a pretty good match. Yes, his volatility is better than mine, but I strongly suspect that's because he's responding faster. The advantage that, I believe, my method has is faster response to changing asset returns - for example, I don't want to use a 30 year average return for bonds if they've been in a 30 year bull market... I'd rather move my expectations with the recent past to accomodate changes.

SD Window and Correlation Window

Initially, I was simply using the same length of time to calculate the SD and the correlation, which is what BTM insinuated was sufficient (i.e. around 20 - 50 days was enough for both).

My results agree with this, but with an interesting refinement. Below is how the returns / Sharpe is affected by varying the SD and correlation window together (i.e. each are equal in length).

Image

Notice there is a rapid increase as this window is increased, peaking in this case at 23 days. However, there is a second peak out towards the long-term - in this case, 813 days!

I thought that was pretty interesting and decided to try varying the SD and correlation windows independently. It turns out that the second peak is caused by the extended correlation window, not the SD.

Keeping the SD window short (23 days), but varying the correlation window independently reveals that longer periods are superior and that the second peak in the previous plot is entirely due to the "better" correlation calculations:

Image

Rebalance Frequency

I've updated a previous post, so check that for my results and opinions.

Summary: shorter = better, just like BTM says. Anything shorter than 80 days is "good enough" if you're lagging the market, as I am.

Gold Cap

Because I'm using a moving arithmetic return for my model, the swings in asset weights can be huge. I noticed that BTM's model never went more than 32% in GLD over the entire period, so I thought about implementing a limit manually. The risk in doing this of course is that you're basically placing more constraints on the model, such that you "break" its asset weights.

This (rapid change in asset weights) is partly mitigated by the reduced rebalance frequency I'm using, but can also be manually overridden by using a cap on the maximum allocation to gold. In general, I don't use this cap, but the code provides it and essentially reallocates anything above the gold cap into the other assets according to the same rules.

Here's an example done using the assets I wish to use, and using a moving arithmetic return window, rebalance frequency of 12 days and a gold cap of 20%:

Image

Here's the same done with no limit on how much to weight towards gold:

Image

You can clearly see GLD can quite happily go wherever it wants with no limit...

(Incidentally, the CAGR over this period was 9% without a cap on gold and 8.2% with a cap of 20%, but I realise that the dataset is too short to know what is "correct". I don't think I'm happy about applying artificial caps in this way, but I know that some people would be uncomfortable with so much in "a barbarous relic" ;) )

Leverage

Since I'm lagging the market and trading less frequently, my results with leverage are very hit and miss. As BTM has commented in a previous post, leverage requires a faster response time to be effective, so I'm not going to use it. In my testing, using a 12 day rebalance window and a 1 day lag, leveraged "BTM" beats buy n hold, which is nice, but loses out to a non-leveraged BTM because it cannot respond fast enough to nasty corrections (e.g. stocks in 2000, 2008 and gold in 2011/12).

Putting it all together

So, in summary, I'm now in a place where I have code that replicates what BTM has outlined, but tweaked for my personal beliefs and situation.

The settings I plan to use are:

1) 23 day SD window.
2) ~4 year arithmetic return window.
3) ~4 year correlation window.
4) 12 day rebalance frequency (I might increase this to 20, 40 or even 80, but I want to "play" and testing suggests that shorter is better).
5) Always use yesterday's close (largely mitigated by a reduced rebalance frequency).

Testing this on ~40 years of data, but only for three assets (Gold, Cash, Stocks) gives me the following:

10.9% CAGR
0.731 Sharpe

Compared with a rebalanced equal split (33/33/33):

5.5% CAGR
0.695 Sharpe

Or 60/40 Stocks/Gold:

8% CAGR
0.658 Sharpe

Over the 40 years of data, my implementation of BTM's method beats a rebalanced 60/40 over 80% of the time.



Testing this on ~13 years and the same assets as BTM (GLD in USD, TLT, S&P):

10.5% CAGR
0.731 Sharpe

vs. an equal split (basically a rebalanced Permanent Portfolio):

6.1% CAGR
0.934 Sharpe

vs. a 60/20/20 rebalanced split of S&P/Bonds/Gold:

6.2% CAGR
0.718 Sharpe

Again, the BTM method beats a 60/20/20 split more often than not (61% in this short dataset). My Sharpe is never as good as BTM's, but as outlined above, that's because I'm trading on a lag and less frequently.


Testing this on my actual assets (a very short dataset of only the last 4 years):

9% CAGR
1.275 Sharpe

vs. equal split:

7.3%
1.443 Sharpe

vs. 60/20/20:

10.4% CAGR
1.683 Sharpe

Pretty happy with that. The data is short enough that a fixed ratio could beat my method due to chance.

Let's go and do this for real...
You're doing a great job exploring the portfolio theory. You've poking into places I haven't as well. The usefulness of really long term correlation is interesting.

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

Post by breakingthemarkt » Sun Feb 23, 2020 5:48 pm

watchnerd wrote:
Sun Feb 23, 2020 10:38 am
jimbomahoney wrote:
Sun Feb 23, 2020 7:09 am


I believe that BTM is also using this for real, and has stated as much.
But for how long?

The posts only go back to March, 2019.

The public record isn't even a year old.

I've asked for sharing of the records going back prior to that. Nothing shared yet.

As you yourself just said, backtesting can prove pretty much any algorithm works if you pick the right data set. The real world tends to pan out differently.
I'd provide further trading histories, but moved my accounts around a just over a year ago so I could trade multiple accounts at once (IRAs and after tax money at the same time). I think the older trades are gone as the accounts are closed. But, even if I did still have them, I don't have a decade+ of history, which is all is all I suspect would make you happy.

Backtesting can prove anything. You can hunt for the perfect inputs to make your test look amazing But I'd like to point out, I showed my history, Jim Showed his, and Chi showed his backtest. They were all good. I have no idea who they are and can tell we are not using the same inputs. However, we are all using the same concepts in the portfolio construction. Those concepts are not arbitrary. They are the theoretical best way to compound wealth over time.

The fact that we all produced strong results with different inputs in the same framework should demonstrate that the framework might have merit.

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

Post by jimbomahoney » Thu Feb 27, 2020 3:44 am

Uncorrelated wrote:
Sun Feb 23, 2020 4:13 pm
If you believe that market timing is impossible, then using a window equal to the entire dataset is the best possible approach. This is the most unbiased estimator possible to sample the true underlying mean of the return distribution.

I don't find your reasons to be convincing. If a limited window is better than an estimate based on the full dataset, then the only explanations I can come up with are chance and momentum.
You will be pleased to know that I now concede defeat on this point.

I've been getting some worrying variance in the returns depending on the window I choose.

I was previously convinced that having a long enough dataset would remove any bias from a specific window, but I get more reliable returns and a better Sharpe if I simply calculate a "fixed" arithmetic return for as much data as possible. I will try and reassure myself that this fixed value will continue to change, albeit very slowly, as time moves and the dataset is filled with current values.

Sometimes a moving window produces higher returns, sometimes much higher, but sometimes it does worse than a "fixed" value over as much data as possible.

I've updated my posts to reflect that and will continue onwards with a non-moving window for arithmetic returns.

Thanks for your challenging discussion! I've once again been talked out of trying to time the market! :sharebeer

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

Post by jimbomahoney » Fri Feb 28, 2020 9:38 am

OK, some more testing and fine-tuning results...

Having been convinced that applying a rolling window to determine the arithmetic returns is prone to bias, (thanks in no small part to Uncorrelated's persuasive techniques and in small part by my previous negative experiences with market-timing!) I'm now using a "fixed" value for the arithmetic return calculated over the entire peiod of the data available.

Testing against BTM's results means that's ~18 years, or~40 years if I use only GLD and GSPC.

Here's an update of my method, which now almost exactly matches BTM's method, but with the following settings:

25 Day Rolling window for SD
45 Day Rolling window for Correlation
Arithmetic Return for the whole period
1 day lag (since I cannot / will not trade in "real-time")
Rebalanced daily
36% Cap on gold allocation (not necessary here as my calculations never go higher than ~20% using a fixed arithmetic return window)

Image

That is pretty damn sweet!

Even with 1 day lag, I almost match his returns and the SD for the returns is also almost as good as his.

I've also done some more testing on the effects of the SD / Correlation / Rebalance Frequency, which are slightly different when using a fixed arithmetic return.

In summary, my thoughts / results are that shorter SD / Correlation / Rebalance periods are better, however, when lagging the market, as I will be, the Rebalance period should be longer than the correlation period, otherwise it mitigates the benefits from taking the correlation into account.

As with any backtesting, longer rebalance periods also result in a smaller sample size (there are less rebalances to test), and therefore prone to noise.

On the data I plan to actually trade with, this means that the sweet spot is somewhere around the middle, so I will be rebalancing every 100 days, which means I get at least two correlation measurements in that time. Either way, the rebalance period doesn't have huge effects when lagging the market by a day - the lowest annual return is 8.3% and the highest is 9.6% (for the markets and time period I'm using - the 9.6% return is a one-off and only had 7 rebalances, which means it's probably a fluke). Everything from 85 - 130 days rebalance returns ~9%.

I've now balanced my portfolio accordingly - the next rebalance date will be ~8th May. Like a true Boglehead, I'm ignoring the current girations, safe in the knowledge that I have a mix of cash (very little), gold, bonds and stocks, as well as property, on which I'm paying down the mortgage at the slowest possible rate because the interest is so low. Should the interest rate on the mortgage increase beyond my annual returns, I will of course pay that down in preference.

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

Post by StochasticEfficiency » Fri Feb 28, 2020 11:25 am

This is fantastic thread Jim, thank you for starting it. Great job on trying to poke holes / test and experiment with what BTM has done. I have tried and tested what BTM has done myself -- and would like to ask how you handle negative weights when the optimal weights are derived i.e. how you impose the no-shorting constraint when some of the assets at times (esp since vol is dynamic and the estimation of expected return is a lot more static) will fall out of the mixing range. Would like to hear your thoughts about this .. or if this is not an issue that you are facing -- then I have to reconsider what I have been doing! :P

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

Post by klaus14 » Fri Feb 28, 2020 3:15 pm

jimbomahoney wrote:
Fri Feb 28, 2020 9:38 am
OK, some more testing and fine-tuning results...

Having been convinced that applying a rolling window to determine the arithmetic returns is prone to bias, (thanks in no small part to Uncorrelated's persuasive techniques and in small part by my previous negative experiences with market-timing!) I'm now using a "fixed" value for the arithmetic return calculated over the entire peiod of the data available.

Testing against BTM's results means that's ~18 years, or~40 years if I use only GLD and GSPC.

Here's an update of my method, which now almost exactly matches BTM's method, but with the following settings:

25 Day Rolling window for SD
45 Day Rolling window for Correlation
Arithmetic Return for the whole period
1 day lag (since I cannot / will not trade in "real-time")
Rebalanced daily
36% Cap on gold allocation (not necessary here as my calculations never go higher than ~20% using a fixed arithmetic return window)

Image

That is pretty damn sweet!

Even with 1 day lag, I almost match his returns and the SD for the returns is also almost as good as his.

I've also done some more testing on the effects of the SD / Correlation / Rebalance Frequency, which are slightly different when using a fixed arithmetic return.

In summary, my thoughts / results are that shorter SD / Correlation / Rebalance periods are better, however, when lagging the market, as I will be, the Rebalance period should be longer than the correlation period, otherwise it mitigates the benefits from taking the correlation into account.

As with any backtesting, longer rebalance periods also result in a smaller sample size (there are less rebalances to test), and therefore prone to noise.

On the data I plan to actually trade with, this means that the sweet spot is somewhere around the middle, so I will be rebalancing every 100 days, which means I get at least two correlation measurements in that time. Either way, the rebalance period doesn't have huge effects when lagging the market by a day - the lowest annual return is 8.3% and the highest is 9.6% (for the markets and time period I'm using - the 9.6% return is a one-off and only had 7 rebalances, which means it's probably a fluke). Everything from 85 - 130 days rebalance returns ~9%.

I've now balanced my portfolio accordingly - the next rebalance date will be ~8th May. Like a true Boglehead, I'm ignoring the current girations, safe in the knowledge that I have a mix of cash (very little), gold, bonds and stocks, as well as property, on which I'm paying down the mortgage at the slowest possible rate because the interest is so low. Should the interest rate on the mortgage increase beyond my annual returns, I will of course pay that down in preference.
nice work.
there is something i am wondering but i don't have the coding setup to test it.
how would it look like if we could incorporate these ideas to a volatility targeting framework.
so we construct the portfolio with:

Given last mont's std and correlations and given return expectations: what is the best mix of these 4 assets to maximize return with the constraint: volatility under 10%?

i suspect this kind of maximization approach will have better grounding than ad hoc adjustments.
35% US, 20 ExUS Dev, 10% EM, 10% EM Bonds, 10% Gold, 10% EDV, 5% I/EE Bonds.

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

Post by Uncorrelated » Fri Feb 28, 2020 4:21 pm

klaus14 wrote:
Fri Feb 28, 2020 3:15 pm
how would it look like if we could incorporate these ideas to a volatility targeting framework.
Volatility targeting is a special case of mean-variance optimization which assumes that return is positive linear with volatility. Because this trading strategy already uses mean-variance optimization (the kelly criterion), volatility targeting is redundant.

The assumptions behind volatility targeting are really stupid. It should never be used.

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

Post by jimbomahoney » Sat Feb 29, 2020 5:09 am

StochasticEfficiency wrote:
Fri Feb 28, 2020 11:25 am
This is fantastic thread Jim, thank you for starting it. Great job on trying to poke holes / test and experiment with what BTM has done. I have tried and tested what BTM has done myself -- and would like to ask how you handle negative weights when the optimal weights are derived i.e. how you impose the no-shorting constraint when some of the assets at times (esp since vol is dynamic and the estimation of expected return is a lot more static) will fall out of the mixing range. Would like to hear your thoughts about this .. or if this is not an issue that you are facing -- then I have to reconsider what I have been doing! :P
Basically, I clip negatives to zero.

In R, the function I found to do that is:

Code: Select all

# Function to clip negative values to zero
ClipToZero <- function(x){ 
  x[x<0] <- 0; x 
}
I do that immediately after calculating the geometric returns (arithmetic - SD^2/2).

I also do it again at the end of the mixing range stage to avoid shorting like you say.

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

Post by ldyrland » Sat Apr 18, 2020 6:32 pm

I'm still not sure how to handle the weights with more than 2 assets. I'm hoping that BTW will give guidance one day on his blog.

I have built a monte carlo simulator and from that, I have produced log normal distributed daily returns for 3 fictitious assets. In addition, I have used a Cholesky process to produce daily return figures that fit any desired correlation between the assets.

I could show all the statistics for my example (avg daily returns, correlations, std deviations, etc), but I don't think that's necessary. I think it's sufficient just to say that when combining just 2 assets at a time, consistent between both my results as well as the formula BTW has in his footnotes on one daily blog, I get the following optimized allocations.

Asset 1 (42.1%) and Asset 2 (57.9%) portfolio
Asset 2 (59.8%) and Asset 3 (40.2%) portfolio
Asset 3 (55.4%) and Asset 1 (44.6%) portfolio

Example. If optimizing a portfolio just using asset 1 and asset 2, then the optimal return is using 42.1% of your portfolio on Asset 1, and 57.9% on Asset 2. So on, and so forth.

I have also determined, through brute force, the optimal allocation of a portfolio using all 3 assets. I'll defer the answer for a little bit. I was hoping Jim, BTW, or anyone else could indicate to me, how they would determine what allocation weights they would assign to each and all 3 assets in this particular example.

Hoping this thread isn't dead anymore.

Lance

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

Post by Hydromod » Sun Apr 19, 2020 9:34 pm

ldyrland wrote:
Sat Apr 18, 2020 6:32 pm
I'm still not sure how to handle the weights with more than 2 assets. I'm hoping that BTW will give guidance one day on his blog.

I have also determined, through brute force, the optimal allocation of a portfolio using all 3 assets. I'll defer the answer for a little bit. I was hoping Jim, BTW, or anyone else could indicate to me, how they would determine what allocation weights they would assign to each and all 3 assets in this particular example.

Hoping this thread isn't dead anymore.

Lance
I'm still waiting for the multiasset algorithm too.

I coded up the 2-asset algorithm in google sheets to compare with other risk parity approaches. I look at three lookback periods (currently 4, 8, and 12 weeks) to compare. My interest is with 3x leveraged ETFs a la Hedgefundie. Generally I try to keep the allocation in the middle of the pack, essentially running dozens of different risk parity tactical allocation schemes at the same time.

I'm not sure exactly how to implement the risk premium for 3x equities (e.g., UPRO or TQQQ) versus 3x treasuries (TMF). I just plot a range of plausible values. Right now it doesn't make too much difference, the equity fraction only changes by a few percent over the range of 0 to 15 % premium for UPRO.

The 4-week indicator has shifted from ~40% to ~25% for over the last week or two; the 8- and 12-week indicators have stayed put at ~35%. I'm not sure how to interpret this shift. Overall the central tendency for the collection of schemes is around 25 to 30%, depending on the assigned risk budget for equities. Early in February, for comparison, the central tendency was more like 70%, with the risk budget for equities set large because the unemployment index was signalling no problem.

TQQQ is a few percent higher than UPRO in all lookback cases.

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

Post by chem » Tue Apr 21, 2020 10:14 am

Did I miss a link, or are there really >100 posts in this thread, including those from 2 posters who have fully implemented this method, but yet there is no complete algorithm laid out or a link to source code for the implementations?

Seems like a whole lot of debate regarding what's going on could be settled by people posting their code.

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

Post by ldyrland » Thu Apr 23, 2020 3:27 pm

I'd be really appreciative of anyone to explain how to calculate allocation percentages for 3, or even 4 assets. I know how to calculate them for 2 assets (formula given in earlier post in BTM's blog post), but it's not obvious how to extend that to more than 2 assets.

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

Post by Uncorrelated » Thu Apr 23, 2020 4:58 pm

ldyrland wrote:
Thu Apr 23, 2020 3:27 pm
I'd be really appreciative of anyone to explain how to calculate allocation percentages for 3, or even 4 assets. I know how to calculate them for 2 assets (formula given in earlier post in BTM's blog post), but it's not obvious how to extend that to more than 2 assets.
I thought I mentioned this earlier, but if you assume that returns follow a normal distribution, it is quite straightforward to calculate the optimal asset allocation for any number of assets with numeric optimization methods. I personally use scipy.minimize to find the optimal solution for arbitrary numbers of assets and correlations. The goal of the optimization is to maximize the utility under an CRRA assumption, which requires the maximization of the equation r - gamma * 1/sigma^2 where gamma is the assumed risk tolerance (gamma = 3 is a reasonable starting point).

My topic viewtopic.php?f=10&t=293469 details some of the mathy stuff, the formula I use is sightly incorrect, one of the comments contains a link to a paper with correct formula's for lognormal distribution assumptions.


I'm working on a similar project for curiosity's sake. All my attempts to forecast returns have failed (as expected). I've found that naive implementations of volatility forecasting fail to outperform out-of-sample, a result contradicting many papers in this field. However, it appears to be possible to gain superior out-of-sample performance with more advanced forecasting methods (i.e. multiple indicators, machine learning, possibly VIX). Finally, optimizing for the correct asset allocation after transaction costs appears to be an extremely difficult problem. I managed to find an acceptable solution for trading on a monthly timescale with reinforcement learning techniques, but so far all my attempts to find a balance between transaction costs and trading profits on a daily timescale have failed.

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

Post by nigel_ht » Fri Apr 24, 2020 8:27 am

watchnerd wrote:
Sun Feb 23, 2020 2:02 pm
jimbomahoney wrote:
Sun Feb 23, 2020 1:39 pm

Is more money better? Sure it is! That's why we're all on this board. But it's just a game to me, and one I'm willing to play around with and potentially, do worse than buy 'n' hold.

:sharebeer
I'm curious how much of your portfolio % you're investing in this particular style of the game.
Seriously why do you care? It's the internet, they can say whatever they want. And for something like this the amount of skin in the game is immaterial to its efficacy. Instead I find questions like this as attempts to bully the OP and dissuade discussions on the merits of an approach.

My primary issue is that the algorithm is opaque...it's not science until someone else can actually replicate your results independently. That's without trying to reverse engineer based on published graphs.

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

Post by Hydromod » Wed May 06, 2020 4:57 pm

There was an update yesterday for two assets plus cash, and this time he provided a google sheet with the example of TLT, SPY, and cash (3-month treasury).

Basically the idea is to do the two-asset approach with each pair of assets individually, then use the lowest of the nonzero percentages for each risky asset. The claim is that this is the exact Kelly criterion when all assets are uncorrelated. The recommended procedure is to follow the same approach with nonzero correlations, which is reported to be close to the exact criterion when the correlations are not too close to one.

The example sheet shows how most of the numbers are calculated using googlefinance and online lookups, but there are still a couple of numbers plugged in to the arithmetic rates that are not explained. These have a significant influence on the calculated weights, especially the one for SPY.

I'm still a bit unclear on what to plug in for expected returns of leveraged ETFs, like TMF and UPRO.

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

Post by klaus14 » Wed May 06, 2020 5:08 pm

Hydromod wrote:
Wed May 06, 2020 4:57 pm
There was an update yesterday for two assets plus cash, and this time he provided a google sheet with the example of TLT, SPY, and cash (3-month treasury).

Basically the idea is to do the two-asset approach with each pair of assets individually, then use the lowest of the nonzero percentages for each risky asset. The claim is that this is the exact Kelly criterion when all assets are uncorrelated. The recommended procedure is to follow the same approach with nonzero correlations, which is reported to be close to the exact criterion when the correlations are not too close to one.

The example sheet shows how most of the numbers are calculated using googlefinance and online lookups, but there are still a couple of numbers plugged in to the arithmetic rates that are not explained. These have a significant influence on the calculated weights, especially the one for SPY.

I'm still a bit unclear on what to plug in for expected returns of leveraged ETFs, like TMF and UPRO.
I would scale the return expectation of SP500 and LTT and subtract the leverage cost.

I would also appreciate if someone publishes a version that supports 3 assets plus cash. And if i can fill in the return expectations.
35% US, 20 ExUS Dev, 10% EM, 10% EM Bonds, 10% Gold, 10% EDV, 5% I/EE Bonds.

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

Post by Uncorrelated » Thu May 07, 2020 4:33 am

Here is a simple example in python that optimizes for the max geometric return under normally distributed returns. It presumably gives answers that are close enough for other distributional assumptions.

Code: Select all

import scipy.optimize
import numpy as np

# returns in excess of risk free rate
asset_returns = np.array([
    .005,    # cash. 0.5% return per year (assumed borrow costs)
    .05,     # stocks, 5% per year
    .02,     # long term bonds. 2% per year.
    .015,    # total bond market
])

# volatility in excess of risk free rate
asset_volatility = np.array([
    .0,     # cash, no volatility
    .16,    # stocks, 16% volatility
    .10,    # long term bonds, 10% volatility
    .03,    # total bond market, 3% volatility
])

# correlations
stock_ltt_corr = 0
stock_tbm_corr = 0
ltt_tbm_corr = .5
asset_correlations = np.array([
    [1,              0,              0,              0],    # correlations with cash all zero
    [0,              1,              stock_ltt_corr, stock_tbm_corr],
    [0,              stock_ltt_corr, 1,              ltt_tbm_corr],
    [0,              stock_tbm_corr, ltt_tbm_corr,   1],
])

# gamma = 1 maximizes log returns
# gamma = 3 probably more realistic for the average investor
gamma = 1.0


def mean_and_std(asset_weights):
    r = np.dot(asset_weights, asset_returns)
    std = asset_weights * asset_volatility
    std = std.reshape((1, len(std)))
    var = np.matmul(np.matmul(std, asset_correlations), std.transpose())
    return r, var[0][0]**.5


def utility(asset_weights):
    mean, std = mean_and_std(asset_weights)
    return mean - .5 * gamma * std**2

cash_bounds = (-2, 0)
asset_bounds = (0, np.inf)

solution = scipy.optimize.minimize(
    lambda x: -utility(x),
    x0=np.full_like(asset_returns, .5),
    bounds=[cash_bounds] + [asset_bounds] * (len(asset_returns)-1),
    constraints=[
        {'type': 'eq', 'fun': lambda x: x.sum() - 1},   # constrain sum(weights) == 1
    ],
    method='SLSQP',
    tol=1e-20
)

print(solution.message)
print("asset weights: ", solution.x)
print("mean: {:6.2%}, std: {:6.2%}".format(*mean_and_std(solution.x)))
sample output:

Code: Select all

Optimization terminated successfully.
asset weights:  [-2.          1.43427915  0.51399576  1.05172509]
mean:  8.78%, std: 24.07%
The example can easily be customized for arbitrary numbers of assets and arbitrary borrowing/leverage constraints. For more complicated problems, it's possible that scipy.optimize.minimize lands in a local minima.

The return is the arithmetic return, annual. For leveraged funds, simply multi[ly the expected return with the leverage amount and subtract the fund ER and internal borrowing costs. Estimating the parameters on a monthly timescale probably results in higher accuracy.

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

Post by Forester » Thu May 07, 2020 6:35 am

Portfolio Visualizer throws up a similar result to the blog strategy using simple target vol. I'm sure the math is very clever but any mix of stocks & LT bonds did just fine since 1980 :confused

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

Post by klaus14 » Thu May 07, 2020 5:32 pm

Uncorrelated wrote:
Thu May 07, 2020 4:33 am
Here is a simple example in python that optimizes for the max geometric return under normally distributed returns. It presumably gives answers that are close enough for other distributional assumptions.

Code: Select all

import scipy.optimize
import numpy as np

# returns in excess of risk free rate
asset_returns = np.array([
    .005,    # cash. 0.5% return per year (assumed borrow costs)
    .05,     # stocks, 5% per year
    .02,     # long term bonds. 2% per year.
    .015,    # total bond market
])

# volatility in excess of risk free rate
asset_volatility = np.array([
    .0,     # cash, no volatility
    .16,    # stocks, 16% volatility
    .10,    # long term bonds, 10% volatility
    .03,    # total bond market, 3% volatility
])

# correlations
stock_ltt_corr = 0
stock_tbm_corr = 0
ltt_tbm_corr = .5
asset_correlations = np.array([
    [1,              0,              0,              0],    # correlations with cash all zero
    [0,              1,              stock_ltt_corr, stock_tbm_corr],
    [0,              stock_ltt_corr, 1,              ltt_tbm_corr],
    [0,              stock_tbm_corr, ltt_tbm_corr,   1],
])

# gamma = 1 maximizes log returns
# gamma = 3 probably more realistic for the average investor
gamma = 1.0


def mean_and_std(asset_weights):
    r = np.dot(asset_weights, asset_returns)
    std = asset_weights * asset_volatility
    std = std.reshape((1, len(std)))
    var = np.matmul(np.matmul(std, asset_correlations), std.transpose())
    return r, var[0][0]**.5


def utility(asset_weights):
    mean, std = mean_and_std(asset_weights)
    return mean - .5 * gamma * std**2

cash_bounds = (-2, 0)
asset_bounds = (0, np.inf)

solution = scipy.optimize.minimize(
    lambda x: -utility(x),
    x0=np.full_like(asset_returns, .5),
    bounds=[cash_bounds] + [asset_bounds] * (len(asset_returns)-1),
    constraints=[
        {'type': 'eq', 'fun': lambda x: x.sum() - 1},   # constrain sum(weights) == 1
    ],
    method='SLSQP',
    tol=1e-20
)

print(solution.message)
print("asset weights: ", solution.x)
print("mean: {:6.2%}, std: {:6.2%}".format(*mean_and_std(solution.x)))
sample output:

Code: Select all

Optimization terminated successfully.
asset weights:  [-2.          1.43427915  0.51399576  1.05172509]
mean:  8.78%, std: 24.07%
The example can easily be customized for arbitrary numbers of assets and arbitrary borrowing/leverage constraints. For more complicated problems, it's possible that scipy.optimize.minimize lands in a local minima.

The return is the arithmetic return, annual. For leveraged funds, simply multi[ly the expected return with the leverage amount and subtract the fund ER and internal borrowing costs. Estimating the parameters on a monthly timescale probably results in higher accuracy.
thanks a lot!
-2 for cash means leverage i guess?
35% US, 20 ExUS Dev, 10% EM, 10% EM Bonds, 10% Gold, 10% EDV, 5% I/EE Bonds.

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

Post by fatFIRE » Fri May 08, 2020 3:05 am

Interesting... do people believe this is legit?

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

Post by nigel_ht » Fri May 08, 2020 6:21 am

fatFIRE wrote:
Fri May 08, 2020 3:05 am
Interesting... do people believe this is legit?
Not being sarcastic: you’re a smart guy...take a look. For me it was too annoying to try to reverse engineer what he did across a bunch of vague blog posts.

I always raise an eyebrow tho when someone claims they solved stuff that has escaped Nobel prize winners in the field. Having worked with a couple they put their pants in like everyone else but they do tend to be pretty sharp...

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

Post by jimbomahoney » Fri May 22, 2020 12:53 pm

I've updated my opening post to say that I'm no longer following this method.

It's a long story, but I'll summarise:
  • I've realised that my attempt to code the "BTM Method" was only an approximation. The latest blog post here made me realise how difficult it is to handle three assets.
  • What I did manage to build was realistic in that it accounted for trading costs and not trading in real-time (e.g. ensuring the model didn't know the day's close until the day after - I believe BTM trades in real-time, which I'm not willing to do).
  • An "equal split" i.e. 1/3 risk assets and zero cash does pretty well and partly explains the results (9.44% CAGR since 1978 on Portfolio Visualiser).
  • I worked with someone who has a lot of experience in the equities market and we really tested the thing to death, using rolling periods to gather 1000s of 3/5/10 year periods. We just couldn't make it work convincingly. Our biggest concern was that volatility doesn't seem a realible indicator of future performance.
That's not to say that Geometric Rebalancing doesn't work, it's just that I can't get it to work.

I will continue to follow the blog and perhaps attempt to code the method once all the details on how to do so are revealed. Many of the building blocks are there.

Until then, I'm back to fixed asset allocations and monthly rebalancing.

In fairness, a lot of positive things came out from the discussions I had with the person I worked with and I learned a lot of new code. Plus, my portfolio has done pretty well, simply due to the fact that it's a mix of uncorrelated assets. I'm up 1.3% YTD and barely noticed the crash - I was down less than 5% at the lows.

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

Post by Dudley » Fri May 22, 2020 1:33 pm

Yes, my take its on aggregate the Permanent Portfolio with a big layer of mumbo-jumbo.

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

Post by imak » Wed May 27, 2020 3:25 pm

I find BTM approach quite similar to Adaptive Asset Allocation which takes into account the volatility, correlation and momentum of N assets and targets volatility based on risk profile. Check out the book "Adaptive Asset Allocation" by Butler, Philbrick & Gordillo.

Both BTM and AAA portfolio rebalancing approaches are quite attractive for post-retirement withdrawal phase as they exhibit much higher perpetual and safe withdrawal rates historically.
Asset Allocation: 30% FNDX, 30% FNDA, 10% FNDF, 10% FNDC, 10% USRT, 10% EDV; Discipline matters more than allocation ~ W Bernstein

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

Post by snailderby » Wed May 27, 2020 6:59 pm

jimbomahoney wrote:
Fri May 22, 2020 12:53 pm
I've updated my opening post to say that I'm no longer following this method.

It's a long story, but I'll summarise:
  • I've realised that my attempt to code the "BTM Method" was only an approximation. The latest blog post here made me realise how difficult it is to handle three assets.
  • What I did manage to build was realistic in that it accounted for trading costs and not trading in real-time (e.g. ensuring the model didn't know the day's close until the day after - I believe BTM trades in real-time, which I'm not willing to do).
  • An "equal split" i.e. 1/3 risk assets and zero cash does pretty well and partly explains the results (9.44% CAGR since 1978 on Portfolio Visualiser).
  • I worked with someone who has a lot of experience in the equities market and we really tested the thing to death, using rolling periods to gather 1000s of 3/5/10 year periods. We just couldn't make it work convincingly. Our biggest concern was that volatility doesn't seem a realible indicator of future performance.
That's not to say that Geometric Rebalancing doesn't work, it's just that I can't get it to work.

I will continue to follow the blog and perhaps attempt to code the method once all the details on how to do so are revealed. Many of the building blocks are there.

Until then, I'm back to fixed asset allocations and monthly rebalancing.

In fairness, a lot of positive things came out from the discussions I had with the person I worked with and I learned a lot of new code. Plus, my portfolio has done pretty well, simply due to the fact that it's a mix of uncorrelated assets. I'm up 1.3% YTD and barely noticed the crash - I was down less than 5% at the lows.
Thanks for the update! So is your current strategy 1/3 U.S. stocks, 1/3 long treasuries, and 1/3 gold?

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

Post by jimbomahoney » Sat May 30, 2020 5:10 am

snailderby wrote:
Wed May 27, 2020 6:59 pm
Thanks for the update! So is your current strategy 1/3 U.S. stocks, 1/3 long treasuries, and 1/3 gold?
No, but that is very tempting.

It's very similar to the Golden Butterfly - i.e. a bias towards equities.

For various reasons, I've gone for 55% Global Stock Market Index (inc. Emerging), 25% Gold, 17% Long Bonds, 3% Cash.

i.e. just enough cash for one year's expenses (of course right now, 3% of cash is nowhere near that level, but I'm in the accumulation phase).

Today's post at BTM is interesting, as it allows us to compare the performance of the Permanent, 1/3 split, 1/3 split with some cash (7%) as well as my chosen asset weights for YTD.

First, here's BTM:

Image

Now, here's a more fair comparison (notice BTM only includes stocks) with other, similar, but fixed-weight, portfolios:

Image

Thirds / Thirds Plus Cash gives over 10% YTD.
Permanent is 10%
My Weights are about 6%.
BTM is 5%.
S&P is -7%.

This simulation was run in R using a simple monthly rebalance rule (in this case, the rebalance happens to be around the 19th of each month. It also includes trading costs as a % of the assets, with gold being more expensive to trade.

Granted, this is over a very particular period of time, and I'm not suggesting that any of these fixed-weight allocations will beat BTM over the long term. According to his historical data, the method returns something like 12% CAGR.

The problem I have is that I cannot trust that BTM's historic returns are realistic - i.e. whether they include trade costs and whether they are avoiding look-ahead bias. When backtesting, because the dates are all aligned in one simple table, it's very easy to rebalance or trade on, say, the 20th based on the close the same day. In reality, you wouldn't know the close that day until it was too late, so you'd need to wait until the 21st.

BTM says he trades in real-time, just before the close, which is perhaps what provides some (all?) of the benefit, but I'm still dubious that recent historical correlation and volatility predicts future performance. I ran lots of tests to see if that was the case, and it wasn't convincing.

My gut feeling at present is that, like most financial trading rules, it's mathematically attractive and therefore easy to justify why it "works", but in reality, the real reason it "works" is simply because it's a Permanent Portfolio with almost no cash.

If it truly works (I'm 50/50 on this), I suspect it may only work if traded in real-time - i.e. shortly before the close.

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

Post by Hydromod » Sat May 30, 2020 10:22 am

jimbomahoney wrote:
Sat May 30, 2020 5:10 am
If it truly works (I'm 50/50 on this), I suspect it may only work if traded in real-time - i.e. shortly before the close.
The issue of trading time is considered in the recent Allocate Smartly post Adding a 1-day lag when executing taa strategies. They looked at the effect of a one-day lag between signal and execution for tactical asset allocation strategies in the period of 1990 through 4/2020. The results suggest that the one-day lag had a negligible effect on strategy performance except at the end of the month (i.e., positive and negative outcomes balance out), although they show a plot suggesting that the effect has varied over the decades.

They point to the end of the month as being a significant exception, and show a plot suggesting that the first and last days of the month have traded differently. I read this as consistent with end-of-the-month trading focused on shorter horizons than the rest of the month (e.g., meeting monthly targets).

Presumably the timing during the day would have an equally negligible effect for weekly trades; lags might become more significant if the trading strategy executed multiple times a day.

This would suggest that the real-time trading aspect is not the dominant factor in the BTM portfolio performance.

get_g0ing
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Has breakingthemarket.com been analyzed?

Post by get_g0ing » Sat May 30, 2020 5:49 pm

[Thread merged into here, see below. --admin LadyGeek]

Hi guys,

I came across this portfolio website https://breakingthemarket.com

He is using 4 assets: Stock Bond Gold Cash
It seems very similar to the permanent portfolio.

Has anyone looked at and analyzed this portfolio strategy?

Thanks.

ChrisBenn
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Re: Has breakingthemarket.com been analyzed?

Post by ChrisBenn » Sat May 30, 2020 5:53 pm


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

Post by LadyGeek » Sat May 30, 2020 6:25 pm

I merged get_g0ing's thread into the on-going discussion.
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