non-intuitive but simple math: the sensitivity of asset allocation

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random_walker_77
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non-intuitive but simple math: the sensitivity of asset allocation

Post by random_walker_77 »

The math of rebalancing is straightforward. So when simple math is surprising, I think it's worth pointing out. One of the things that once surprised me was that the sensitivity of the asset allocation ratio is much less than intuition might suggest.

Let's say you have a 60/40 stock/bond allocation. If stocks *double*, with no gain in bonds, what's your allocation become?
Answer: 75/25
(i.e. before: 60K/100K, after: 120K/160K)

Over time, bonds will probably grow, so you'd likely be closer to 70/30.

Here's a table showing what happens to the allocation ratio, assuming bonds are unchanged, when stocks double or halve. I also calculated how much of a stock increase is needed to raise the allocation of stocks by 10%

Code: Select all

                                                                        
Orig alloc         after 100% stock growth      After 50% drop          stock growth to reach next 10% threshold
stocks  bonds           stocks  bonds           stocks  bonds                           historical return, 1926-2020**
30      70              46.2    53.8            17.6    82.4            55.6%           7.7%
40      60              57.1    42.9            25.0    75.0            50.0%           8.1%
50      50              66.7    33.3            33.3    66.7            50.0%           8.6%
60      40              75.0    25.0            42.9    57.1            55.6%           9.0%
70      30              82.4    17.6            53.8    46.2            71.4%           9.4%
80      20              88.9    11.1            66.7    33.3            125.0%          9.7%
90      10              94.7    5.3             81.8    18.2            inf             10.0%
100     0               100.0   0.0             100.0   0.0             n/a             10.2%

** historical return from https://investor.vanguard.com/investing/how-to-invest/model-portfolio-allocation
As an engineer that's pretty comfortable with math, I was surprised that I was surprised. Sharing this in case it's helpful to others as well.

This suggests that if you set a rebalancing strategy where you rebalance when stock allocation has drifted by an absolute 10%, your stocks have likely grown by over 50% (which isn't necessarily a bad thing, mind you, but it is good to understand).
dbr
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by dbr »

Yes, it is surprising how infrequently market values move enough to actually dictate a need to rebalance.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by nisiprius »

Yep. It's been noted before, notably by The Finance Buff who published this chart in 2011. But it deserves to be mentioned more often, to counteract the mystique in the folkways which overemphasizes the need, importance, and benefits of rebalancing.

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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by bertilak »

dbr wrote: Sun Nov 28, 2021 12:08 pm Yes, it is surprising how infrequently market values move enough to actually dictate a need to rebalance.
It seems to me that rebalancing concerns on this board have declined considerably. Do I simply not notice because I mentally filter out those topics or have they actually decreased?

Perhaps I used to pay more attention. I remember it being analyzed to death, with discussions about the proper way to determine when to rebalance: by time (occasionally, on one's birthday), by band (absolute vs. relative), and by amount (all the way back to the target or just back to within the band). These possibilities were debated, back tested and Monte Carlo tested.

My conclusion about rebalancing, in the investment how-to I wrote for my wife, was "It doesn't much matter."
May neither drought nor rain nor blizzard disturb the joy juice in your gizzard. -- Squire Omar Barker (aka S.O.B.), the Cowboy Poet
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by ThankYouJack »

bertilak wrote: Sun Nov 28, 2021 1:42 pm
dbr wrote: Sun Nov 28, 2021 12:08 pm Yes, it is surprising how infrequently market values move enough to actually dictate a need to rebalance.
It seems to me that rebalancing concerns on this board have declined considerably. Do I simply not notice because I mentally filter out those topics or have they actually decreased?

Perhaps I used to pay more attention. I remember it being analyzed to death, with discussions about the proper way to determine when to rebalance: by time (occasionally, on one's birthday), by band (absolute vs. relative), and by amount (all the way back to the target or just back to within the band). These possibilities were debated, back tested and Monte Carlo tested.

My conclusion about rebalancing, in the investment how-to I wrote for my wife, was "It doesn't much matter."
I’ve noticed similar and have similar thoughts about rebalancing. I rebalance at most once a year.

What I find ironic though is RBD is still going strong.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by bertilak »

ThankYouJack wrote: Sun Nov 28, 2021 1:52 pm
bertilak wrote: Sun Nov 28, 2021 1:42 pm
dbr wrote: Sun Nov 28, 2021 12:08 pm Yes, it is surprising how infrequently market values move enough to actually dictate a need to rebalance.
It seems to me that rebalancing concerns on this board have declined considerably. Do I simply not notice because I mentally filter out those topics or have they actually decreased?

Perhaps I used to pay more attention. I remember it being analyzed to death, with discussions about the proper way to determine when to rebalance: by time (occasionally, on one's birthday), by band (absolute vs. relative), and by amount (all the way back to the target or just back to within the band). These possibilities were debated, back tested and Monte Carlo tested.

My conclusion about rebalancing, in the investment how-to I wrote for my wife, was "It doesn't much matter."
I’ve noticed similar and have similar thoughts about rebalancing. I rebalance at most once a year.

What I find ironic though is RBD is still going strong.
Instead of "rebalancing" I talk about "maintaining" the AA by judicious buying and selling during normal activity: RMDs, gifting, expenses. An RMD is both selling (obviously) and buying (reinvesting). I also take dividends to the settlement fund and occasionally buy from there. If I was still contributing (I'm now retired) it would be even easier.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by pizzy »

bertilak wrote: Sun Nov 28, 2021 2:06 pm
ThankYouJack wrote: Sun Nov 28, 2021 1:52 pm
bertilak wrote: Sun Nov 28, 2021 1:42 pm
dbr wrote: Sun Nov 28, 2021 12:08 pm Yes, it is surprising how infrequently market values move enough to actually dictate a need to rebalance.
It seems to me that rebalancing concerns on this board have declined considerably. Do I simply not notice because I mentally filter out those topics or have they actually decreased?

Perhaps I used to pay more attention. I remember it being analyzed to death, with discussions about the proper way to determine when to rebalance: by time (occasionally, on one's birthday), by band (absolute vs. relative), and by amount (all the way back to the target or just back to within the band). These possibilities were debated, back tested and Monte Carlo tested.

My conclusion about rebalancing, in the investment how-to I wrote for my wife, was "It doesn't much matter."
I’ve noticed similar and have similar thoughts about rebalancing. I rebalance at most once a year.

What I find ironic though is RBD is still going strong.
Instead of "rebalancing" I talk about "maintaining" the AA by judicious buying and selling during normal activity: RMDs, gifting, expenses. An RMD is both selling (obviously) and buying (reinvesting). I also take dividends to the settlement fund and occasionally buy from there. If I was still contributing (I'm now retired) it would be even easier.
To this point, how do people handle contributions? Do you always direct non-automated contributions towards underweighted asset classes? Or do you invest contributions at target allocation and then rebalance at other times?
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by delamer »

We have 5 classes in our asset allocation.

So there is a lot more interplay of the rates of return for those 5 classes relative to those for just 2 classes, in terms of maintaining our target allocation.

It also means that being 4 percentage-points (for example) off target for a given asset is more significant than if we had a 60/40 portfolio and were off by the same 4 points for each category.
One thing that humbles me deeply is to see that human genius has its limits while human stupidity does not. - Alexandre Dumas, fils
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by pizzy »

delamer wrote: Sun Nov 28, 2021 2:24 pm We have 5 classes in our asset allocation.

So there is a lot more interplay of the rates of return for those 5 classes relative to those for just 2 classes, in terms of maintaining our target allocation.

It also means that being 4 percentage-points (for example) off target for a given asset is more significant than if we had a 60/40 portfolio and were off by the same 4 points for each category.
Isn’t this the crux of the 5/25 rebalancing rule?
Last edited by pizzy on Sun Nov 28, 2021 6:45 pm, edited 1 time in total.
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delamer
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by delamer »

pizzy wrote: Sun Nov 28, 2021 2:32 pm
delamer wrote: Sun Nov 28, 2021 2:24 pm We have 5 classes in our asset allocation.

So there is a lot more interplay of the rates of return for those 5 classes relative to those for just 2 classes, in terms of maintaining our target allocation.

It also means that being 4 percentage-points (for example) off target for a given asset is more significant than if we had a 60/40 portfolio and were off by the same 4 points for each category.
Isn’t this the crux of the 5/20 rebalancing rule?
What is that rule — 5 percentage points or 20 percent?
One thing that humbles me deeply is to see that human genius has its limits while human stupidity does not. - Alexandre Dumas, fils
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by Beensabu »

delamer wrote: Sun Nov 28, 2021 2:45 pm
pizzy wrote: Sun Nov 28, 2021 2:32 pm
delamer wrote: Sun Nov 28, 2021 2:24 pm We have 5 classes in our asset allocation.

So there is a lot more interplay of the rates of return for those 5 classes relative to those for just 2 classes, in terms of maintaining our target allocation.

It also means that being 4 percentage-points (for example) off target for a given asset is more significant than if we had a 60/40 portfolio and were off by the same 4 points for each category.
Isn’t this the crux of the 5/20 rebalancing rule?
What is that rule — 5 percentage points or 20 percent?
It's 5% for stock:bond AA and 25% for sub asset classes.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by neurosphere »

I find that as balances grow, small percentage drops in stocks "feel" worse due to the absolute dollar amount, even though percentage asset allocation (and thus "risk") may not have changed materially. Having an asset allocation target and rebalancing makes it "feel" like I'm doing something "good" for my financial future. Thus, I think having a particular rebalancing strategy (annually? every Olympics?) is still prudent.

Also, I "believe" (i.e. have no idea how robust the data is) there is a rebalancing bonus due to the general concept of reversion to the mean (in the lay sense).
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by SantaClaraSurfer »

It's also non-intuitive at the other end of the telescope.

When you rebalance consistently with new money, any individual contribution can look very different from your nominal AA.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by Karamatsu »

I find that as balances grow, small percentage drops in stocks "feel" worse due to the absolute dollar amount, even though percentage asset allocation (and thus "risk") may not have changed materially.
This is important and quite real. Percentages tend to mislead. If you reframe a loss as "How many years did I work in order to save up that much money?" or "How many years would have I have to work and save to make up the loss?" It becomes much more real, and clarifies, at least for me, how the way we talk about these things can seduce people into taking far more risk than they're really able to handle or absorb. This goes 10X when people are in actual retirement, for whom the seemingly easy decision to work more/longer is less of an option.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by delamer »

Beensabu wrote: Sun Nov 28, 2021 2:55 pm
delamer wrote: Sun Nov 28, 2021 2:45 pm
pizzy wrote: Sun Nov 28, 2021 2:32 pm
delamer wrote: Sun Nov 28, 2021 2:24 pm We have 5 classes in our asset allocation.

So there is a lot more interplay of the rates of return for those 5 classes relative to those for just 2 classes, in terms of maintaining our target allocation.

It also means that being 4 percentage-points (for example) off target for a given asset is more significant than if we had a 60/40 portfolio and were off by the same 4 points for each category.
Isn’t this the crux of the 5/20 rebalancing rule?
What is that rule — 5 percentage points or 20 percent?
It's 5% for stock:bond AA and 25% for sub asset classes.
25% is too high for me, but to each their own.
One thing that humbles me deeply is to see that human genius has its limits while human stupidity does not. - Alexandre Dumas, fils
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by Mike Scott »

I think some of the rebalancing discussion has been centered around whether there is a "rebalancing bonus" and, if so, how to catch it. My take away from many discussions is that it seems either very small or unlikely and rebalancing is more about risk management.

I think the wide (5%) rebalancing bands better reflect the lack of precision in asset allocation rather than the exact ratios usually stated. For me to state that I have a 75/25 AA does not give the same impression as stating the allowed range of a 75/25 (+/-5) AA. It also leaves out the complexities of exactly what makes up those two numbers.

Some of if it is convenience in simplifying concepts and typing and some of it is the false sense of precision that we may want to impose on our investments when we really don't have that much control.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by pseudoiterative »

There's another way that asset allocation can be very sensitive.

Suppose you're trying to use mean variance optimisation to compute what your "optimal" asset allocation should be, from input data consisting of estimated future returns, variance, covariance across all the different asset classes, and your personal risk-reward preference. Thinking of the optimisation process as a function from input data that produces an output asset allocation, this optimisation process can be highly sensitive to the inputs, and may not even be continuous. There might be multiple asset allocations that are all estimated to give comparably "optimal" results in terms of expected risk-adjusted return, but those allocations themselves might look very dissimilar. If you observe new data that gives you a slightly more up to date estimate of covariance between two asset classes, it's possible a tiny change in this statistic causes a large change in which asset allocation is found to be most "optimal".

Of course, if you actually tried to run a procedure like this every day to spit out the latest estimate of the "optimal" asset allocation given the current estimated conditions, and tried to rebalance to it, you'd burn lots of your money on trading fees, buy/sell spreads and taxes.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by Kobinator »

pizzy wrote: Sun Nov 28, 2021 2:15 pm To this point, how do people handle contributions? Do you always direct non-automated contributions towards underweighted asset classes? Or do you invest contributions at target allocation and then rebalance at other times?
My preference: When contributions are frequent, I'd rather contribute at the target and rebalance when I hit a trigger. Mainly because I don't want to spend the time/effort to adjust contributions every paycheck. When contributions are rare, it makes more sense to go through the effort of calculating the current allocation and balancing it out.
Mike Scott wrote: Sun Nov 28, 2021 3:38 pm I think some of the rebalancing discussion has been centered around whether there is a "rebalancing bonus" and, if so, how to catch it. My take away from many discussions is that it seems either very small or unlikely and rebalancing is more about risk management.
The main benefit to me is giving my higher-order brain a feeling of doing something/managing to the conditions during a crisis. Otherwise, my lizard brain would like to run for the hills and I may sell out, haha.

The graph above also applies to stock run-ups. And the mental benefit is good then, too, but more about exercising the will to follow a plan rather than let "winners run".
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by Beensabu »

delamer wrote: Sun Nov 28, 2021 3:32 pm
Beensabu wrote: Sun Nov 28, 2021 2:55 pm
delamer wrote: Sun Nov 28, 2021 2:45 pm
pizzy wrote: Sun Nov 28, 2021 2:32 pm
delamer wrote: Sun Nov 28, 2021 2:24 pm We have 5 classes in our asset allocation.

So there is a lot more interplay of the rates of return for those 5 classes relative to those for just 2 classes, in terms of maintaining our target allocation.

It also means that being 4 percentage-points (for example) off target for a given asset is more significant than if we had a 60/40 portfolio and were off by the same 4 points for each category.
Isn’t this the crux of the 5/20 rebalancing rule?
What is that rule — 5 percentage points or 20 percent?
It's 5% for stock:bond AA and 25% for sub asset classes.
25% is too high for me, but to each their own.
I think it's 25% of the % allocated to the sub asset class. So if it was 20% of overall, then a +/- 5% in that allocation to overall would hit the rebalancing band.
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by delamer »

Beensabu wrote: Sun Nov 28, 2021 4:11 pm
delamer wrote: Sun Nov 28, 2021 3:32 pm
Beensabu wrote: Sun Nov 28, 2021 2:55 pm
delamer wrote: Sun Nov 28, 2021 2:45 pm
pizzy wrote: Sun Nov 28, 2021 2:32 pm

Isn’t this the crux of the 5/20 rebalancing rule?
What is that rule — 5 percentage points or 20 percent?
It's 5% for stock:bond AA and 25% for sub asset classes.
25% is too high for me, but to each their own.
I think it's 25% of the % allocated to the sub asset class. So if it was 20% of overall, then a +/- 5% in that allocation to overall would hit the rebalancing band.
Say the desired allocation to a sub-asset was 20% (of total), with a 25% rebalancing trigger. Then if sub-asset allocation fell to less than 15% or more than 25% (again, of the total), rebalancing would be in order.

That’s my interpretation, although I’d never heard the “rule” so I could be mistaken.

I’m not sure if we are agreeing, or not.
One thing that humbles me deeply is to see that human genius has its limits while human stupidity does not. - Alexandre Dumas, fils
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Re: non-intuitive but simple math: the sensitivity of asset allocation

Post by Beensabu »

delamer wrote: Sun Nov 28, 2021 5:40 pm
Beensabu wrote: Sun Nov 28, 2021 4:11 pm
delamer wrote: Sun Nov 28, 2021 3:32 pm
Beensabu wrote: Sun Nov 28, 2021 2:55 pm
delamer wrote: Sun Nov 28, 2021 2:45 pm

What is that rule — 5 percentage points or 20 percent?
It's 5% for stock:bond AA and 25% for sub asset classes.
25% is too high for me, but to each their own.
I think it's 25% of the % allocated to the sub asset class. So if it was 20% of overall, then a +/- 5% in that allocation to overall would hit the rebalancing band.
Say the desired allocation to a sub-asset was 20% (of total), with a 25% rebalancing trigger. Then if sub-asset allocation fell to less than 15% or more than 25% (again, of the total), rebalancing would be in order.

That’s my interpretation, although I’d never heard the “rule” so I could be mistaken.

I’m not sure if we are agreeing, or not.
That's my interpretation as well. It's a Larry thing, I think,

I dunno. I just rebalance all the things sort of when I decide to rebalance. It only happens in the big account. I don't care so much.
"The only thing that makes life possible is permanent, intolerable uncertainty; not knowing what comes next." ~Ursula LeGuin
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