Backtesting with "factored" rebalance bands

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Pet Hog
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Backtesting with "factored" rebalance bands

Post by Pet Hog » Fri Jan 30, 2015 8:10 pm

In May 2013, my first post on this forum considered 40/15 rebalance bands as an alternative to 35/15 or 40/10.  My argument remains that 40/15 bands better reflect the doubling or halving of a PP component and that they catch an even number of rebalances on the downside and upside.  My limited testing at the time showed, however, that there was no gain in CAGR when using 40/15 bands, but I still think the idea of "symmetrically factored" rebalance bands is valid.  My Excel-fu has increased considerably lately, allowing me to test a wider range of "factors," not just 40/15.  Here are the results.  I used daily data from peaktotrough.com.  Consider the following chart of CAGRs of PPs begun on 1/1/1972 and maintained until yesterday (1/29/2015) with rebalancing occurring in response to a factor, plotted on the x-axis:

[img width=500]http://s12.postimg.org/azll7vupp/CAGR_factored.png[/img]

I am defining the factor as the amount the PP component (e.g., stocks) grew or shrank; a factor of 2 means that the component doubled or halved in value.  So a portfolio of 25/25/25/25 would become 50/25/25/25 or 12.5/25/25/25.  In the former, the component that grew has become 40% of the overall portfolio; in the latter it has become 14.29%.  Thus, for a factor of 2, the rebalance bands are 40/14.29.  Traditional rebalancing at 35/15 corresponds to an upside factor of 1.62 and a downside factor of 1.89 -- that's why I don't consider it "symmetrical."

Since 1/1/1972, a PP with 35/15 rebalancing bands and dividends reinvested has returned a CAGR of 9.29%.  That's the benchmark.  I wondered whether there was a factored rebalancing strategy that would have beaten this value; the results are in the chart above, which displays the CAGRs and number of rebalancing events for factors ranging from 1 (in essence, rebalancing every day) to 13 (no rebalancing during the 43 years of backtesting).

There are several interesting features in the chart.
  • There was a strategy that would have returned an 11.06% CAGR, albeit with only two rebalances.  That factor was 12.93, representing bands of 81.17 and 2.51% -- quite extreme!
  • A strategy that would have resulted in rebalancing on average every three years (from 13 to 15 times in 43 years) would have had factors from 1.65 (35.48/16.81 bands) to 1.98 (39.76/14.41 bands), but with quite a swing in CAGRs, from a minimum of 9.03% to a maximum of 9.63%.  Thus, small changes in rebalance band percentages can have dramatic effects on returns.
  • A factor of 3.60 (54.55/8.47 bands) gave a CAGR of 9.71% (4 rebalances), yet a tiny increase in the factor to 3.61 plunged the CAGR down to just 8.91% (2 rebalances).  I think that so few rebalances led to data mining effects that skewed the results, which depend on this one particular starting date.
  • A factor of 12.97 (81.21/2.51 bands) led to no rebalancing ever; the CAGR was 8.26%.  This CAGR is the lowest in the whole chart, meaning that any one of these factored rebalancing strategies defeated the set-and-forget portfolio.  In other words, there is definitely a rebalancing bonus.
  • It isn't easy finding a factor that would guarantee beating the CAGR of 9.29% for the 35/15 portfolio. Perhaps somewhere between 3.0 (50/10 bands) and 3.5 (53.85/8.70 bands), with just four rebalances, but the drop at 3.6 loses all of those gains.  Any factor above 5.5 (64.71/5.71 bands) seems to beat it, but would any of us be happy rebalancing only twice in 43 years?  By the way, I don't think there is anything magical about 35/15.  I have performed a similar analysis of 25+x/25-x bands and found just as much noise and sensitivity.  I'll post those results soon.
My conclusion is that 35/15 is a great strategy for getting a decent CAGR.  I haven't determined maximum drawdowns, but, if you are risk-tolerant, a set-and-forget portfolio with very wide bands (71.18/4.30 at a factor of 7.41) would have given a CAGR of 10% over the last 43 years; it remains quite balanced -- here's its chart:

[img width=500]http://s28.postimg.org/6f45h3frx/741_fa ... _value.png[/img]
Last edited by Pet Hog on Sun Feb 01, 2015 5:12 pm, edited 1 time in total.
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buddtholomew
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Re: Backtesting with "factored" rebalance bands

Post by buddtholomew » Fri Jan 30, 2015 8:22 pm

Wonderful, but I think there is a mistake with your no-rebalancing ever option (set and forget). Can you confirm an un-rebalanced portfolio only refurned 8.26 CAGR and 81.17/2.51 returned the highest value? Seems contrary to what I have always considered factual - un-rebalanced equals highest return, more volatility.
"The first principle is that you must not fool yourself and you are the easiest person to fool" --Feynman.
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Re: Backtesting with "factored" rebalance bands

Post by Pet Hog » Fri Jan 30, 2015 10:41 pm

Hi Buddtholemew

If you check http://www.peaktotrough.com/hbpp.cgi and put 1/1/1972 as the starting date, rebalance as "none," and reinvest dividends as "yes," you will get a CAGR of 8.24%.  That matches my value.  I agree, unbalanced equals more volatility, and with the 81.17/2.51 bands you are pretty much setting-and-forgetting, too, so there is also much volatility.  But you would have captured two extreme multiyear momentum effects.  It's not obvious from the lower chart, but the last rebalance with the 71.18/4.30 bands was probably some time in 1995 (presumably because gold got very low).  So it's been a 20-year set-and-forget since then.  The only other rebalance was in 1979 or 1980, that time because gold got too high.

Actually, I am very surprised by the results because I was trying to find the optimal approach, thinking it would be around 35/15, and indeed when I tested the factors up to 2.0, it seemed that was the case, but with no clear winning strategy.  Only when, for a bit of fun, I looked at extreme cases I saw that some random factors gave >10% returns.  It makes sense to me that if stock, bond, and gold bull/bear cycles last 10, 15, or 20 years, then the optimal rebalancing strategy will be one matching those absolute peaks and troughs, rather than their temporary, local ones.

It occurred to me that, with such infrequent rebalancing, maybe just being 100% in stocks or bonds or gold might have been a better strategy because they have all done well since 1972, but that's not the case, either.  They have returned just 9.54, 8.44, and 8.04% respectively.  So there is definitely a benefit to diversifying among these volatile assets and rebalancing only in extreme events.
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Re: Backtesting with "factored" rebalance bands

Post by buddtholomew » Sat Jan 31, 2015 6:19 pm

Thank you for the explanation and the analysis Pet Hog.
"The first principle is that you must not fool yourself and you are the easiest person to fool" --Feynman.
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Re: Backtesting with "factored" rebalance bands

Post by MachineGhost » Sat Jan 31, 2015 9:13 pm

Actually, I am very surprised by the results because I was trying to find the optimal approach, thinking it would be around 35/15, and indeed when I tested the factors up to 2.0, it seemed that was the case, but with no clear winning strategy.  Only when, for a bit of fun, I looked at extreme cases I saw that some random factors gave >10% returns.  It makes sense to me that if stock, bond, and gold bull/bear cycles last 10, 15, or 20 years, then the optimal rebalancing strategy will be one matching those absolute peaks and troughs, rather than their temporary, local ones.

It occurred to me that, with such infrequent rebalancing, maybe just being 100% in stocks or bonds or gold might have been a better strategy because they have all done well since 1972, but that's not the case, either.  They have returned just 9.54, 8.44, and 8.04% respectively.  So there is definitely a benefit to diversifying among these volatile assets and rebalancing only in extreme events.
Are you're looking for symmetrical 41.67%/15% rebalancing bands so that losses are covered to breakeven back at the 25% point?

You can do 100% into any of the four assets via tactical allocation, but the risk will be higher than the PP.  There's enough of a lag in switching before a new asset can pick up the slack.  Some people could be willing to pay that price for a higher return.  I'm not one of them.  However, it is more robust than DecisionMoose.
"All generous minds have a horror of what are commonly called 'Facts'. They are the brute beasts of the intellectual domain." -- Thomas Hobbes

Disclaimer: I am not a broker, dealer, investment advisor, physician, theologian or prophet.  I should not be considered as legally permitted to render such advice!
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Re: Backtesting with "factored" rebalance bands

Post by Pet Hog » Sun Feb 01, 2015 2:17 pm

MachineGhost wrote:Are you're looking for symmetrical 41.67%/15% rebalancing bands so that losses are covered to breakeven back at the 25% point?
MG, the traditional bands for the PP (30/20, 35/15, 40/10) don't capture "buy low" as well as they do "sell high."  These "25 + x" and "25 – x" bands when taken to their extreme (x = 25) would require rebalancing at 50/0, meaning you would never rebalance because something is on sale!  The effect is more subtle for, say, 35/15, where rebalances still tend to favor one asset reaching 35% before another reaches 15%.  I was wondering whether symmetrically factored bands would provide a better return by evening up the number of rebalances caused by breaches on the low and high sides, but I'm not sure they do.  I think there's something advantageous to profit taking (selling high).  I have some more modeling to do, but I wonder what would happen if the lower band were ignored completely and rebalancing were to occur only if, say, 35% were breached.
MachineGhost wrote:You can do 100% into any of the four assets via tactical allocation, but the risk will be higher than the PP.  There's enough of a lag in switching before a new asset can pick up the slack.  Some people could be willing to pay that price for a higher return.  I'm not one of them.  However, it is more robust than DecisionMoose.
I'm not one of them either.  There are obviously more complicated strategies for timing the market than just using the PP's rebalance bands.  I'm just trying to find what would have been the best rebalance band strategy.  35/15 remains hard to beat!
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Re: Backtesting with "factored" rebalance bands

Post by Kbg » Sun Feb 01, 2015 5:39 pm

Too often we forget the primary purpose of rebalancing...to actually track the target asset allocation more closely.

Methodologically it is probably better to spend time understanding the four (or whatever) assets and if they deliver when and how they are supposed to deliver under the conditions by which the asset allocation was constructed. Once that has been established, a study of correlations over time and historical volatility of the assets is a fruitful field of study to refine one's understanding of the interrelationships between them. Then, if one feels they have a good dial on/ability to predict macro conditions they have established the proper foundation to select an over/under weighting of the assets based on their forecast. Tactically one should also consider tax implications.

Here's a great little chart for contemplation visually.

https://www.bullionvault.com/US_Annual_ ... 5-2014.pdf

What I find most interesting is the how the relationship between gold and government bonds has changed from the left side to the right side of the chart. Clearly inflation/deflation changes the correlation from not correlated to correlated. One I plan on looking into more.
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Re: Backtesting with "factored" rebalance bands

Post by MachineGhost » Sun Feb 01, 2015 6:01 pm

Pet Hog wrote: MG, the traditional bands for the PP (30/20, 35/15, 40/10) don't capture "buy low" as well as they do "sell high."  These "25 + x" and "25 – x" bands when taken to their extreme (x = 25) would require rebalancing at 50/0, meaning you would never rebalance because something is on sale!  The effect is more subtle for, say, 35/15, where rebalances still tend to favor one asset reaching 35% before another reaches 15%.  I was wondering whether symmetrically factored bands would provide a better return by evening up the number of rebalances caused by breaches on the low and high sides, but I'm not sure they do.  I think there's something advantageous to profit taking (selling high).  I have some more modeling to do, but I wonder what would happen if the lower band were ignored completely and rebalancing were to occur only if, say, 35% were breached.
I'm still struggling to wrap my head around this, but I think I'm getting the ideal.  But one thing you should be cognizant of is the wildy differing volatilities of the assets at any point in time.  I don't think any firm conclusions can be drawn unless they are always risk normalized before hitting a rebalancing band.  A relatively more volatile asset will hit the rebalancing bands more often than a relatively less volatile assset.  I think cash will never hit an upper or lower rebalancing band because it has no volatility, although I'm probably wrong about that in some extreme situations.

Why don't you experiment with custom rebalancing bands for each of the four assets?  Or tie a factor to the asset's volatility.

If it wasn't for the negative correlations of the assets as sophie pointed out, buying low would be an utter disaster as it typically is when done to assets in isolation.

BTW, for a 10% loss it would actually be 11% on the upside from 25%, 36% not 41.67% as I suggested.  i.e. 36%/10% bands are symmetrical.
Last edited by MachineGhost on Sun Feb 01, 2015 6:25 pm, edited 1 time in total.
"All generous minds have a horror of what are commonly called 'Facts'. They are the brute beasts of the intellectual domain." -- Thomas Hobbes

Disclaimer: I am not a broker, dealer, investment advisor, physician, theologian or prophet.  I should not be considered as legally permitted to render such advice!
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Re: Backtesting with "factored" rebalance bands

Post by Pet Hog » Sun Feb 01, 2015 6:08 pm

Here is a chart of the CAGRs of PPs started on 1/1/1972 and ended on 1/29/2015 (the same date range as above, using the same daily data).  The rebalance bands are "traditional" equally ranged ones (35/15, 30/20, 40/10, and so on), but I have modeled all ranges (25 ± x) from ±0.1% (rebalance almost daily) to ±57% (never rebalance because bands are too wide at 82/0).

[img width=500]http://s4.postimg.org/uu8cnmpj1/CAGRs_e ... _bands.png[/img]
  • Again, all of these rebalance strategies were better than never rebalancing.
  • Very wide bands (±43%, so 68/0 bands) gave CAGRs greater than 10%, but now with only one rebalance over 43 years!
  • Maximum CAGR was 11.39% with ±55.9% range (80.9/0 bands).
  • It's hard to find a range that would provide a CAGR consistently greater than that provided by 35/15 bands.
Zooming in on the bands from 30/20 to 50/0:

[img width=500]http://s4.postimg.org/3kwz94ofx/expande ... ranged.png[/img]

35/15 (the red dot, 9.32% CAGR) looks like a good strategy, but there is a lot of noise in the region.  For example, a slight increase in the range to ±10.2% (35.2/14.8 bands) gave a much better CAGR (9.62%), whereas a slight decrease in the range to ±9.4% (34.4/25.6 bands) gave a slightly poorer CAGR (9.14%).  In fact, I suggest that the bump in CAGR observed around the 35/15 bands is an artifact, much like the dip at ±14 (39/11 bands); they probably arise from good or bad luck in the timing of some of their rebalances.  If you ignore those two blips, the trend channel is pretty consistent, sloping upward, for ranges from ±5 to ±16.  I would have to model some other periods, maybe 30-year periods, with different start dates to see whether some of the noise can be eliminated and some definite conclusions can be drawn.  I still think 35/15 is a great strategy.
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Re: Backtesting with "factored" rebalance bands

Post by craigr » Sun Feb 01, 2015 10:10 pm

A primary reason to rebalance is to control risk as well as keep returns a lot less volatile. So although there are definite ways to move the bands around to boost the returns, ultimately we need to balance this against investor behavior patterns that tend to flee from risk and make bad decisions during market drops. The 35/15 rebalancing bands for me represent a pretty good balance between taking on too much risk and not enough.

Mostly as we explained in the book, it's about having firewalls in your portfolio. You always want to own enough of an asset that it can help when needed, but not so much that you take a serious pounding if the floor drops out from under it. The firewall idea works with the 25% split because even if an asset drops 50% tomorrow morning and no assets go up, the most damage to the portfolio would be -12.5%. This is not great, but a heck of a lot better than other strategies taking huge concentrated bets and the volatility with them. Or even against someone not rebalancing at all and finding they have, say, 50% in one particular asset and the others a lot less.

I found the 15/35 or 20/30 rebalancing bands (or thereabouts in between if you are doing things for tax reasons), work pretty well as you've found. They are also simple for most people to track and execute on without getting too wound up over percentages.

Thanks for the interesting data.
Last edited by craigr on Sun Feb 01, 2015 10:12 pm, edited 1 time in total.
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