## "There is a rebalancing bonus ... false"

Discuss all general (i.e. non-personal) investing questions and issues, investing news, and theory.
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### Re: "There is a rebalancing bonus ... FALSE"

rmelvey wrote:Rebalancing brings your portfolio's geometric return closer to the weighted average arithmetic return of its components. Momentum need not have anything to do with it. It is a function of diversification reducing steep losses, which we all know kills compounding.

I made a post awhile ago that shows it using an extreme example devised by Claude Shannon:
http://www.stableinvesting.com/2013/04/ ... demon.html

This example is interesting because obviously there is no momentum. Also there is no "buy low" and "sell high" because valuations have nothing to do with random coin flips. The real takeaway is that the individual coin flip games have a positive expected arithmetic return, but a an expected geometric return of 0. However, diversification can help close the gap between the arithmetic and geometric returns.

So, who is trading with Shannon? There must be another bettor taking the "tails" side of the proposition. The "tails" bettor must also be making money at the same rate if he rebalances according to Shannon's logic. But, somebody has to be losing. Even if you extend it to a large pool of bettor/rebalancers (some betting "head" and others betting "tails"), it still cannot work. Shannon is somehow assuming an idealized source of money that just does not exists in the stock market where aggregate behavior of bettors set prices. The problem with Shannon's mythical stock market is:

(1) Events (coin flips) are impacting prices with no buying or selling whatsoever, whereas in the real world events cause investors to buy and sell and that buying and selling sets prices.

(2) All buying and selling is rebalancing and it's having no impact on prices, whereas in the real world rebalancing would be the only thing that impacted prices if all buying and selling was rebalancing.

bmdaniel
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### Re: "There is a rebalancing bonus ... false"

LadyGeek wrote:^^^ I got it working in LibreOffice Calc (8.2 MB). It still takes a long time to update (40 S), but I can see the principles involved.

A minor comment: Formulas in the Outcomes tab depend on the cell to the right. Cells in the last column (GU) refer to Column GV, which is blank. Blank cells are interpreted as 0, which creates a bias for Column GU.

I don't think it's significant and wouldn't belabor the point, I just wanted to mention it.

I think the GV you are seeing is linking to the Coin Flip tab (where it's not blank, the columns just don't match exactly b/c of the starting column on the Outcomes tab). If you look down the GU column on the outcomes tab, it's a blend of wins and losses.

bmdaniel
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### Re: "There is a rebalancing bonus ... FALSE"

rmelvey wrote:Rebalancing brings your portfolio's geometric return closer to the weighted average arithmetic return of its components. Momentum need not have anything to do with it. It is a function of diversification reducing steep losses, which we all know kills compounding.

I made a post awhile ago that shows it using an extreme example devised by Claude Shannon:
http://www.stableinvesting.com/2013/04/ ... demon.html

This example is interesting because obviously there is no momentum. Also there is no "buy low" and "sell high" because valuations have nothing to do with random coin flips. The real takeaway is that the individual coin flip games have a positive expected arithmetic return, but a an expected geometric return of 0. However, diversification can help close the gap between the arithmetic and geometric returns.

So, who is trading with Shannon? There must be another bettor taking the "tails" side of the proposition. The "tails" bettor must also be making money at the same rate if he rebalances according to Shannon's logic. But, somebody has to be losing. Even if you extend it to a large pool of bettor/rebalancers (some betting "head" and others betting "tails"), it still cannot work. Shannon is somehow assuming an idealized source of money that just does not exists in the stock market where aggregate behavior of bettors set prices. The problem with Shannon's mythical stock market is:

(1) Events (coin flips) are impacting prices with no buying or selling whatsoever, whereas in the real world events cause investors to buy and sell and that buying and selling sets prices.

(2) All buying and selling is rebalancing and it's having no impact on prices, whereas in the real world rebalancing would be the only thing that impacted prices if all buying and selling was rebalancing.

The you can't have a tails better taking the other side of the proposition; he would have the opposite payout structure, and a negative expected value (i.e., when it's heads he loses his whole amount, but when it's tails he only gets half his amount). This is one of the biggest issues with applying his criterion directly to investing (it only applies when you can estimate the positive expected value and total risk of a "bet") - if you don't know that you have a positive EV, you shouldn't bet at all.

swaption
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### Re: "There is a rebalancing bonus ... FALSE"

rmelvey wrote:Rebalancing brings your portfolio's geometric return closer to the weighted average arithmetic return of its components. Momentum need not have anything to do with it. It is a function of diversification reducing steep losses, which we all know kills compounding.

I made a post awhile ago that shows it using an extreme example devised by Claude Shannon:
http://www.stableinvesting.com/2013/04/ ... demon.html

This example is interesting because obviously there is no momentum. Also there is no "buy low" and "sell high" because valuations have nothing to do with random coin flips. The real takeaway is that the individual coin flip games have a positive expected arithmetic return, but a an expected geometric return of 0. However, diversification can help close the gap between the arithmetic and geometric returns.

So, who is trading with Shannon? There must be another bettor taking the "tails" side of the proposition. The "tails" bettor must also be making money at the same rate if he rebalances according to Shannon's logic. But, somebody has to be losing. Even if you extend it to a large pool of bettor/rebalancers (some betting "head" and others betting "tails"), it still cannot work. Shannon is somehow assuming an idealized source of money that just does not exists in the stock market where aggregate behavior of bettors set prices. The problem with Shannon's mythical stock market is:

(1) Events (coin flips) are impacting prices with no buying or selling whatsoever, whereas in the real world events cause investors to buy and sell and that buying and selling sets prices.

(2) All buying and selling is rebalancing and it's having no impact on prices, whereas in the real world rebalancing would be the only thing that impacted prices if all buying and selling was rebalancing.

You know, this hypothetical example seemingly ends up in a dead end when applied ot the real world. Or does it? On the surface, this geometric 100% gain and 50% loss does not really exist, at least in the world of coin flips. Even if one flip, it's a bad bet for the house and no such thing exists in Vegas or anywhere. Vegas is a world with no expected return. But of course Kelly and Shannon were dealing with a different kind of world. They were dealing in a world where there was some sort of advantage, something that gave an expectation of a return. The Kelly Criteria was essentially an approach to sizing bets so that the advantage could be realized without going bust.

But once again, how does this apply to the real world of investing where it is presumed there is nobody on the other side of these bad bets? Equities are different. There actually is an expectation of a return over the long term in the form of the risk premium. That's why the coin flip model presented here is actually somewhat appropriate. But once again, who is on the other side of the table? In other words, who is the house? In the case of equities, it is those that raise the capital in the first place, essentially the corporation. In the case of trading, it is whoever sells the stocks to you in that they are foregoing an expectation of positive return.

In this regard, the Kelly Criteria can be applicable to portfolio management. The presumed size of the risk premium could govern the size of the bet, essentially shifting funds to equity when the advantage increases in the form of a higher premium. To some extent I guess this is rebalancing, or perhaps it may actually be tactical asset management.

technovelist
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### Re: "There is a rebalancing bonus ... FALSE"

staythecourse wrote:How many times do we have to go over very simple concepts?? Rebalancing is done to MANTAIN the same risk as your static allocation. Of course, the risk would go up you rebalanced into a risky asset class. NO duh!! The fact remains is OVER time there is NO way a riskier asset class will underperform a safe asset class. That is a against the fundamentals of the relationship between risk and return.

Any extra return is based on diversification return, i.e. risk/ return of the portfolio is greater then the weighted calculation of its components. Now that folks can argue away.

Good luck.

So you are saying that riskier asset classes don't have more risk if you just wait long enough? I don't think that is correct.
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nisiprius
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### Re: "There is a rebalancing bonus ... false"

The reason this stuff isn't very applicable to the real world is that in the real world nobody tells you what the underlying probabilities are. You don't know how much your wealth gets multiplied for every head or divided for every tail. You don't know the mean of the distribution from which stock prices are drawn. You can be pretty darn certain that whatever the numbers you are looking at are, they are not independent samples. You don't know how many parameters are inside the black box. You don't know that they don't change. If they change, you don't know when they change or how often they change.

You don't know whether the stock market is the same thing before and after the Federal Reserve Act, or splitting of stocks to make them affordable to the general public in the 1920s, or the Investment Company Act of 1940, or the end of fixed brokerage commissions, or decimalization, or the rise of the 401(k), or HFT. You don't know whether apparent differences at different times, like the decline of the "small company effect," are because of sampling error or because something changed.

All the "probabilities" are not probabilities at all, they are estimates deduced from observation. An observer of the pink curve who applies the most sophisticated statistical analysis to decide the optimum investing strategy is going to come to the wrong conclusion, unless they have a valid model for what's inside the black box.

To see caption, highlight it by dragging mouse across the text between these lines
Growth of \$1 invested in Bernard Madoff's Fairfield Sentry
To see caption, highlight it by dragging mouse across the text between these lines
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swaption
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### Re: "There is a rebalancing bonus ... false"

nisiprius wrote:The reason this stuff isn't very applicable to the real world is that in the real world nobody tells you what the underlying probabilities are. You don't know how much your wealth gets multiplied for every head or divided for every tail. You don't know the mean of the distribution from which stock prices are drawn. You can be pretty darn certain that whatever the numbers you are looking at are, they are not independent samples. You don't know how many parameters are inside the black box. You don't know that they don't change. If they change, you don't know when they change or how often they change.

You don't know whether the stock market is the same thing before and after the Federal Reserve Act, or splitting of stocks to make them affordable to the general public in the 1920s, or the Investment Company Act of 1940, or the end of fixed brokerage commissions, or decimalization, or the rise of the 401(k), or HFT. You don't know whether apparent differences at different times, like the decline of the "small company effect," are because of sampling error or because something changed.

Of course to some extent, you are correct that there are all sorts of uncertainties. But in thinking about all of this, it is helpful to consider the various things that you do know, or at least those things that one typically assumes to be true.

For example, the notion of a geometric return pattern can't be correct. The real world just does not fit into such nice neat models. But we do approach all of this with some expectation of the existence of an equity risk premium, which implies some expectation of assymetry of returns. In other words, one is more likely to win than lose. We also know, or think that we know that the risk premium varies over time. We don't necessarily know whether that is mean reverting, but I think we believe it to be behavioral in nature, without necessarily much insight beyond that.

Constructs such as the Kelly Criteria are intended to deal with these types of situations. The premise is that you have some form of bet where there is an advantage. Counting cards in blackjack is a very simplified example. It is gambling with an expectation of a positive return. The construct varies the size of the bet based on the size of the advantage, and also sets as a priority not going bust. Simply put, if you have an advantage, the priority should be staying in the game for as long as possible.

Equity investors are thought to have an advantage in the form of the risk premium. A construct like the Kelly Criteria can be thought to imply the very sound portfolio management approaches that lead to diversification and bond allocations that minimize volatilty, and enable investors to 'stay in the game'. With this in mind, along the empirical evidence, the existence of a rebalancing bonus should not be easily dismissed.

rmelvey
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### Re: "There is a rebalancing bonus ... false"

nisi,

I think taking the game to it's next logical extreme will help you realize the importance of geometric means. Let's say the game was a 50/50 chance of losing 100% or gaining 1000%. If you knew that you were going to repeat this game forever, would you still bet all of your money? It's pretty clear that at some point you would lose everything if you were salivating over the arithmetic returns, going all in. A geometric mean maximizer would use the Kelly criterion to find the optimal betting size.

The bottom line is that going all in, even if the odds are in your favor, can hazardous to your long run wealth. That is pretty intuitive to the lay person, and the Kelly criterion (focusing on geometric mean maximizing) is the more nerdy way of quantifying that.

nisiprius
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### Re: "There is a rebalancing bonus ... false"

rmelvey, I wish you would state clearly what it is that you think rebalancing accomplishes. Because this thread is about the reality of a rebalancing bonus or lack thereof.

Personally I am risk-averse and I am not not not saying we should "make log of mean wealth big" if "years to act are long." I am all in favor of rebalancing for risk control, I am all in favor of shaping the distribution of outcomes to one in which the bottom tail is quite tolerable, and I believe stocks are risky in the short run, the medium run, and the long run.

But on your web page you certainly seem to be implying that rebalancing does something or other to increase returns.

You say that the game presents an "incredible profit opportunity" exists, but only for the "intelligent player [who wagers] only half of their bankroll each round" and that "This seemingly small differences turns the game into a winner." To say it "turns the game into a winner" means that it was not a winner for the first player.

Well, that's incorrect. The game is an incredible profit opportunity for the player who bets half, and an even more incredible profit opportunity--with more risk--for the player who bets all.

In the example you chose to present, a "rebalancing bonus" only occurs if the number of heads and tails is equal, which, of course, amounts to assuming perfect mean reversion. If heads and tails are random, there is no rebalancing bonus, there is no rebalancing penalty, and the player who bets half as much receives half as large a return per flip (and thus compounds at half the speed).

If you want to say that rebalancing is a good thing, I agree. If you want to say that it is virtuous sort of reverse martingale, and shapes the distribution of outcomes in a way that most investors should prefer to have it shaped, no problem.

The problem is that many investors really believe that rebalancing actually manufactures extra returns all by itself, through buying low and selling high. Well, It doesn't. No mean reversion, no rebalancing bonus. And if you will agree with that, then, no problem at all.
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rmelvey
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### Re: "There is a rebalancing bonus ... false"

nisiprius wrote:rmelvey, I wish you would state clearly what it is that you think rebalancing accomplishes. Because this thread is about the reality of a rebalancing bonus or lack thereof.
But on your web page you certainly seem to be implying that rebalancing does something or other to increase returns.

nisi,

You have to internalize the difference between arithmetic returns and geometric returns. I keep saying it but you keep ignoring it

I agree that mean reversion is necessary for rebalancing to increase average arithmetic returns. However, my example didn't have that at all! In my example, it is very clear to see that rebalancing into 50% cash cut the the arithmetic average return in half. There is no mean reversion in the example in my article. However, you can't eat average arithmetic returns, you eat average geometric returns and rebalancing can most certainly boost those.

Its kind of ironic that people keep saying that rebalancing reduces risk, but does not increase return. Holding arithmetic returns constant, decreasing risk increases the geometric return. Look at the example of the portfolio rebalanced between three coin flips. The arithmetic average return of the portfolio is the same as the arithmetic average return of its components, but rebalancing among the three clearly increases compounded returns. That simulation runs every 10 minutes or, offering a fresh chart each time. If your logic was correct, the long run returns of the portfolio would be the same as the weighted average of its components, but it is clearly not.

In summary, rebalancing increases returns because it decreases risk. Buying low and selling high has absolutely nothing to do with the bonus I am talking about.
Last edited by rmelvey on Tue May 28, 2013 3:29 pm, edited 1 time in total.

rmelvey
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### Re: "There is a rebalancing bonus ... false"

It's very clear to see that it is pure randomness.

YDNAL
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### Re: "There is a rebalancing bonus ... false"

rmelvey wrote:
nisiprius wrote:rmelvey, I wish you would state clearly what it is that you think rebalancing accomplishes. Because this thread is about the reality of a rebalancing bonus or lack thereof. But on your web page you certainly seem to be implying that rebalancing does something or other to increase returns.

nisi,

You have to internalize the difference between arithmetic returns and geometric returns. I keep saying it but you keep ignoring it

Can someone (anyone) please tell me the arithmetic returns (or is it geometric?) that I will get over the next decade (June 1, 2013 - May 31, 2023) ??

I really appreciate it !
Landy | Be yourself, everyone else is already taken -- Oscar Wilde

swaption
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### Re: "There is a rebalancing bonus ... false"

rmelvey wrote:
nisiprius wrote:rmelvey, I wish you would state clearly what it is that you think rebalancing accomplishes. Because this thread is about the reality of a rebalancing bonus or lack thereof.
But on your web page you certainly seem to be implying that rebalancing does something or other to increase returns.

nisi,

You have to internalize the difference between arithmetic returns and geometric returns. I keep saying it but you keep ignoring it

Yes, but people keep rightly pointing out that geometric returns are not an accurate representation, which I keep saying doesn't matter, which up until now you have ignored

What does matter is that there is an expectation of a positive return from equities, which the notion of geometric returns attempts to model. That is what is important. Equities are supposed to be that winning bet where risk aversion (and common sense) dictates that you don't go all in at once. You don't need mean reversion. You just need to be able to place the bet as many times as possible. Rebalancing or the Kelly Criteria do that. If it were mean reverting, then that would just be a stronger case.

RobG
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### Re: "There is a rebalancing bonus ... false"

I've tried to explain this before with moderate success. Part of the problem is jargon... expected return is not what you can "expect" etc. As you guys know, when you combine two uncorrelated assets you get a reduction in standard deviation along with an increase in expected median return (how's that for jargon). Wm Bernstein erroneously referred to this as the "rebalancing bonus" and the world has been confused ever since. I think he finally fessed up to this. You do not need to rebalance get that effect as it is just the stats looking forward from your present position and assumes no future action.

Now if you define the rebalancing bonus as the effect you get by rebalancing (a logical thing to do IMO) it gets a little confusing to say whether it is good or bad (obviously it is"real" because it has *some* effect). If you chose an allocation of 60/40 you did so because that is what you feel optimizes whatever you were trying to optimize, call it warm fuzzy factor (WFF). It might refer to return for a given amount of risk or human's weird desire to have portfolio allocations with nice round numbers. If your allocation drifts to 59/41 you are no longer optimizing your WFF (60/40 by definition) so you need to rebalance. Clearly there is a "rebalancing bonus" because you increased your WFF. Besides what could be worse than having your allocation being the ratio of two prime numbers that add to 100?

If you're carrying bonds rebalancing reduces your return because the highest expected return is 100% stocks, but it will optimize WFF so you are maximally happy lowering your returns! Confused?

If you look at Monte Carlo sims using dependent random variables (NOT historical returns) where you are trying to optimize pure returns there is a itty-bitty improvement if you rebalance. To optimize returns you need to start with two uncorrelated assets with similar return statistics (otherwise the optimum would be to just chose the one with the higher return.. no that isn't exactly true, but bear with me...) For example, if you have two uncorrelated assets, each with a median return of 10% and SD of 20% mixing them 50/50 will improve the stats of your portfolio to 12% median and 14% SD (if memory serves). Now if they change to 49/51 the stats will be less favorable, so again there is a bonus to be had be rebalancing - unfortunately it is near zero on average - like 0.1% if totally uncorrelated, and real assets are correlated.

Furthermore, the optimum might be 49/51 so you could be doing harm by maintaining 50/50.

Confused? Just remember from a theoretical perspective there is no reason to think that rebalancing will add to your returns under any conditions and it will likely subtract since you are maintaining things like bonds. What we do know with 100% certainty is if you try to "wing it" and buy low sell/high you almost always make the wrong decision. In my opinion, that is where the real advantage of a structured rebalancing plan lies.

rg
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### Re: "There is a rebalancing bonus ... false"

From 1972 - 2012 an annually rebalanced portfolio of 50% TSM + 50% 5Yr Treasurys gained 9.28% compounded annually. An un-rebalanced 50/50 portfolio gained 9.00% compounded annually. If the annual cost of rebalancing was less than 0.28%, then you did better by rebalancing; otherwise you didn't. Jack Bogle doesn't think re-balancing is worth messing with unless your allocation target really gets out of line:

We’ve just done a study for the NYTimes on rebalancing, so the subject is fresh in my mind. Fact: a 48%S&P 500, 16% small cap, 16% international, and 20% bond index, over the past 20 years, earned a 9.49% annual return without rebalancing and a 9.71% return if rebalanced annually. That’s worth describing as “noise,” and suggests that formulaic rebalancing with precision is not necessary.

We also did an earlier study of all 25-year periods beginning in 1826 (!), using a 50/50 US stock/bond portfolio, and found that annual rebalancing won in 52% of the 179 periods. Also, it seems to me, noise. Interestingly, failing to rebalance never cost more than about 50 basis points, but when that failure added return, the gains were often in the 200-300 basis point range; i.e., doing nothing has lost small but it has won big. (I’m asking my good right arm, Kevin, to send the detailed data to you.)

My personal conclusion. Rebalancing is a personal choice, not a choice that statistics can validate. There’s certainly nothing the matter with doing it (although I don’t do it myself), but also no reason to slavishly worry about small changes in the equity ratio. Maybe, for example, if your 50% equity position grew to, say, 55% or 60%.

http://seekingalpha.com/article/41119-vanguard-s-jack-bogle-on-rebalancing-don-t
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swaption
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### Re: "There is a rebalancing bonus ... false"

RobG wrote: To optimize returns you need to start with two uncorrelated assets with similar return statistics

But that is not an accurate representation of stock and bonds, at least the similar return part. This is important for reasons explained above.

umfundi
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### Re: "There is a rebalancing bonus ... false"

I tried to gore this ox once before:

viewtopic.php?p=1689566#p1689566

Let me try again. Suppose you invest \$3,000 over three months in a particular fund. Month 1 the price is \$100 per share (10 shares), month 2 it is \$80 (12.5 shares) and month 3 it is \$120 (8.33 shares). Now you have 30.83 shares and your cost basis is \$97.30 per share.

But, wait! The average price per share is \$100. Magic! You have gained 2.7%. The DCA bonus!

Not really. The simple or arithmetic mean (average) is simply the wrong measure. \$97.30 is the geometric (or harmonic) mean and is what applies here.

Let me say this again. The arithmetic average does not apply here. Using it is incorrect.

rmelvey is correct in stating that the simple average is always greater than the harmonic mean. That is an artifact of the arithmetic. It means (pun intended) nothing.

Keith
Last edited by umfundi on Tue May 28, 2013 8:59 pm, edited 1 time in total.
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### Re: "There is a rebalancing bonus ... false"

Here's an oversimplified explanation of the math:

The principle of compounding interest is an exponential function. See: Comparing Investments

IOW, compounding means multiplication (interest on top of interest). Taking an average also needs multiplication. In this case, you multiply the terms together and take the square root of the whole thing. See: Geometric mean

Simple interest means that no multiplication is involved, so you take a "standard" average: Arithmetic mean
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### Re: "There is a rebalancing bonus ... false"

Thorpe generalized Kelly for the stock market:

http://edwardothorp.com/sitebuildercont ... Market.pdf

(Thorpe probably did the same in his book "Beat the Market")

There's a compilation of papers on Kelly Capital Growth Investment Criteria:

http://www.amazon.com/Kelly-Capital-Gro ... 342&sr=1-2

Real-world gambling seems just as messy as the stock market investing relative to thought experiment gambling. The theory of how to beat the Blackjack dealer is pristine, but if you read "Beat the Dealer", you will see that all sorts of casino counter-measures (like cheating dealers who have the slight of hand expertise of a magician) and other implementation shortfall issues make it messy and you have to be an expert in avoiding these to avoid losing your edge.

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### Re: "There is a rebalancing bonus ... false"

As a rough estimate:

geometric mean = arithmetic mean - 1/2 variance

If you have ever noticed, the trailing returns from the stock market are about 10% per year, but the average growth of the stock market in any given year is about 19%. The estimation formula above explains how this can be the case.

It's useful to wrap your mind around this, tends to mitigate one's excitement about a year like we are having right now, more Spock less Bones.

boggler
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### Re: "There is a rebalancing bonus ... false"

geometric mean = arithmetic mean - 1/2 variance

If you have ever noticed, the trailing returns from the stock market are about 10% per year, but the average growth of the stock market in any given year is about 19%. The estimation formula above explains how this can be the case.

It's useful to wrap your mind around this, tends to mitigate one's excitement about a year like we are having right now, more Spock less Bones.

Aren't people here saying that there is a magic way to get closer to the 19 number rather than 10? Why doesn't everyone do this?

swaption
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### Re: "There is a rebalancing bonus ... false"

tadamsmar wrote:Thorpe generalized Kelly for the stock market:

http://edwardothorp.com/sitebuildercont ... Market.pdf

(Thorpe probably did the same in his book "Beat the Market")

There's a compilation of papers on Kelly Capital Growth Investment Criteria:

http://www.amazon.com/Kelly-Capital-Gro ... 342&sr=1-2

Real-world gambling seems just as messy as the stock market investing relative to thought experiment gambling. The theory of how to beat the Blackjack dealer is pristine, but if you read "Beat the Dealer", you will see that all sorts of casino counter-measures (like cheating dealers who have the slight of hand expertise of a magician) and other implementation shortfall issues make it messy and you have to be an expert in avoiding these to avoid losing your edge.

Cool stuff, although clearly not light reading in either case. For anyone at all interested in Kelly, Thorpe, and Shannon, the book Fortune's Formula is a highly recommended read. Includes all of this, yet easily readbable, and genuinely a great story.

http://www.amazon.com/gp/product/0809045990/ref=s9_simh_gw_p14_d0_i2?pf_rd_m=ATVPDKIKX0DER&pf_rd_s=center-2&pf_rd_r=0TD09GXEG6EXCVMXGWR6&pf_rd_t=101&pf_rd_p=1389517282&pf_rd_i=507846

RobG
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### Re: "There is a rebalancing bonus ... false"

swaption wrote:
RobG wrote: To optimize returns you need to start with two uncorrelated assets with similar return statistics

But that is not an accurate representation of stock and bonds, at least the similar return part. This is important for reasons explained above.

That was actually a significant point I was trying to make. That is, only using very optimistic assumptions can you expect an increase in your annualized returns, and even with those assumptions the predicted increase will be very small because it is unusual for your portfolio to get far out of balance with two assets with similar stats. And if they are not similar the portfolio to optimize your returns is made up of mostly the higher performing one so it is even more difficult to drift meaningfully out of allocation over a realistic investment period.

Another example... suppose you've been nailing the recent volatility: you sell off when the market is 2% high, and buy back when it drops by 2%. You've increased your return a whopping 0.04% - and a good portion of the time it won't drop back down 2% so your average gains will be a fraction of 0.04%. It is kind of depressing that the math shows all these ideas really don't increase your returns (unless your portfolio drifts way out of whack), but they certainly increase your WFF and it gives you something to do.

Just my 2 cents.

rg
Stay thrifty my friends.

umfundi
Posts: 3361
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### Re: "There is a rebalancing bonus ... false"

I ran a backtest simulation for the decade of 2000-2010 with a three-fund portfolio and rebalancing. I found that returns were 0.3% per year higher with rebalancing.

That said, it is easy to imagine scenarios (like the market since 2009) where rebalancing might lower your returns.

Keith
Déjà Vu is not a prediction

Posts: 6758
Joined: Mon May 07, 2007 12:33 pm

### Re: "There is a rebalancing bonus ... false"

umfundi wrote:I ran a backtest simulation for the decade of 2000-2010 with a three-fund portfolio and rebalancing. I found that returns were 0.3% per year higher with rebalancing.

That said, it is easy to imagine scenarios (like the market since 2009) where rebalancing might lower your returns.

Keith

There's theory and then there is backtesting. If the market is a random walk, then no bonus. But perhaps the attempted closed loop control of the economy by the Fed et al tends to steer stocks and bonds into oscillations favorable to rebalancing. Either way, some rebalancing at some point is a good idea to reduce the creep toward more risk, I think.

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: "There is a rebalancing bonus ... false"

umfundi wrote:I ran a backtest simulation for the decade of 2000-2010 with a three-fund portfolio and rebalancing. I found that returns were 0.3% per year higher with rebalancing.

That said, it is easy to imagine scenarios (like the market since 2009) where rebalancing might lower your returns.

Keith

There's theory and then there is backtesting. If the market is a random walk, then no bonus. But perhaps the attempted closed loop control of the economy by the Fed et al tends to steer stocks and bonds into oscillations favorable to rebalancing. Either way, some rebalancing at some point is a good idea to reduce the creep toward more risk, I think.

Not quite.

Suppose you have 10 shares of investment A valued at \$100 each, and 10 shares of investment B, valued at \$100 each. Total \$2,000.
Next month, A is worth \$90, B is worth \$110. Total \$2,000.
The month after, A and B return to their original values of \$100 each.

If you did not rebalance, you have \$2,000.

If you did rebalance in the next month, in the end you have 11.111 shares of A and 9.091 shares of B. When A and B return to their original values your investment is worth \$2020.21, a gain of 1%.

Rebalancing gains from the noise. It loses from the trends.

Keith
Déjà Vu is not a prediction

RNJ
Posts: 624
Joined: Mon Apr 08, 2013 9:06 am

### Re: "There is a rebalancing bonus ... false"

Isn't it reasonable to rebalance simply to control one's exposure to risk, without concern for a "bonus"?

RobG
Posts: 1204
Joined: Wed Feb 28, 2007 12:59 am
Location: Bozeman, MT

### Re: "There is a rebalancing bonus ... false"

umfundi wrote:
umfundi wrote:I ran a backtest simulation for the decade of 2000-2010 with a three-fund portfolio and rebalancing. I found that returns were 0.3% per year higher with rebalancing.

That said, it is easy to imagine scenarios (like the market since 2009) where rebalancing might lower your returns.

Keith

There's theory and then there is backtesting. If the market is a random walk, then no bonus. But perhaps the attempted closed loop control of the economy by the Fed et al tends to steer stocks and bonds into oscillations favorable to rebalancing. Either way, some rebalancing at some point is a good idea to reduce the creep toward more risk, I think.

Not quite.

Suppose you have 10 shares of investment A valued at \$100 each, and 10 shares of investment B, valued at \$100 each. Total \$2,000.
Next month, A is worth \$90, B is worth \$110. Total \$2,000.
The month after, A and B return to their original values of \$100 each.

If you did not rebalance, you have \$2,000.

If you did rebalance in the next month, in the end you have 11.111 shares of A and 9.091 shares of B. When A and B return to their original values your investment is worth \$2020.21, a gain of 1%.

Rebalancing gains from the noise. It loses from the trends.

Keith

I guess I would have answered it differently... if it is a random walk there *is* theoretically a bonus that increases the annualized return that you can expect, but it is extremely small (<<0.1%) with the assets we have available - the assets are just too correlated. Even with your example using negatively correlated assets zig-zagging a total of 20% and perfectly timed you only gained 2%.
Stay thrifty my friends.

RobG
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Location: Bozeman, MT

### Re: "There is a rebalancing bonus ... false"

RNJ wrote:Isn't it reasonable to rebalance simply to control one's exposure to risk, without concern for a "bonus"?

I'm just nitpicking... I'd say you are trying to keep it constant. If the market is going up, yes you would be reducing your exposure to risk by selling some equities. But if it is going down you would be increasing your risk by buying more equities.
Last edited by RobG on Thu May 30, 2013 10:42 am, edited 1 time in total.
Stay thrifty my friends.

Browser
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Joined: Wed Sep 05, 2012 4:54 pm

### Re: "There is a rebalancing bonus ... false"

So, has anyone ever figured out whether historically the higher returns from rebalancing (if there are higher returns) is primarily due to the lucky timing of buying stocks when they're down, or to the lucky timing of selling them when they're up? It IS timing in drag you know.
The fox knows many little things, but the hedgehog knows one big thing ~ Archilochus | | "To the man with just a hammer, every problem looks pretty much like a nail" ~ Charlie Munger

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: "There is a rebalancing bonus ... false"

Browser wrote:So, has anyone ever figured out whether historically the higher returns from rebalancing (if there are higher returns) is primarily due to the lucky timing of buying stocks when they're down, or to the lucky timing of selling them when they're up? It IS timing in drag you know.

Browser,

I think I disagree.

How did we get here? You set an asset allocation (AA) to manage your risk. You diversify with the total market to reduce volatility and to get on the efficient frontier. Then, you rebalance to maintain the AA.

Diversification works because different asset classes are not perfectly correlated. One goes up, another goes up less or even goes down. The rebalancing bonus (yes, there is one) comes from this lack of correlation. As I said, it comes from the noise, the relative fluctuations among your diversified investments.

It is not timing. It is maintaining your AA. The bonus is incidental, and may be washed out by a trend:

Invest equally in A and B. In month 1, A goes up 10%, B stays the same. In month 2, A goes up another 10%, B stays the same. Rebalancing loses over not rebalancing, but your AA is drifting to the higher returning (riskier) investment if you do not rebalance.

Personally, I invest predominantly in the Vanguard LifeStrategy Moderate Fund. It rebalances to 60/40 every day. I believe that momentum and reversion to the mean are only visible in your rear view mirror, and that continuous rebalancing is the way to go. I will agree that rebalancing with bands, or on really bad days, is an attempt at programmatic market timing. I reject the idea that automatic rebalancing to maintain an AA, as the LifeStrategy or Target Retirement Funds do, is market timing.

Keith
Déjà Vu is not a prediction

Posts: 6758
Joined: Mon May 07, 2007 12:33 pm

### Re: "There is a rebalancing bonus ... false"

umfundi wrote:
umfundi wrote:I ran a backtest simulation for the decade of 2000-2010 with a three-fund portfolio and rebalancing. I found that returns were 0.3% per year higher with rebalancing.

That said, it is easy to imagine scenarios (like the market since 2009) where rebalancing might lower your returns.

Keith

There's theory and then there is backtesting. If the market is a random walk, then no bonus. But perhaps the attempted closed loop control of the economy by the Fed et al tends to steer stocks and bonds into oscillations favorable to rebalancing. Either way, some rebalancing at some point is a good idea to reduce the creep toward more risk, I think.

Not quite.

Suppose you have 10 shares of investment A valued at \$100 each, and 10 shares of investment B, valued at \$100 each. Total \$2,000.
Next month, A is worth \$90, B is worth \$110. Total \$2,000.
The month after, A and B return to their original values of \$100 each.

If you did not rebalance, you have \$2,000.

If you did rebalance in the next month, in the end you have 11.111 shares of A and 9.091 shares of B. When A and B return to their original values your investment is worth \$2020.21, a gain of 1%.

Rebalancing gains from the noise. It loses from the trends.

Keith

Since the stock market trends over longer terms, reblancing is a loser.

But, you are measuring with only returns and completely ignoring risk. By that measurement you should put it all in risky stocks.

Posts: 6758
Joined: Mon May 07, 2007 12:33 pm

### Re: "There is a rebalancing bonus ... false"

RobG wrote:
RNJ wrote:Isn't it reasonable to rebalance simply to control one's exposure to risk, without concern for a "bonus"?

Nitpicking... I'd say you are trying to keep it constant. If the market is going up, yes you would be reducing your exposure to risk by selling some equities. But if it is going down you would be increasing your risk by buying more equities.

The market price movements are changing risk (or at least the risk proxy of proportion in stocks). You are compensating to get the risk proxy back where you want it.

Browser
Posts: 4504
Joined: Wed Sep 05, 2012 4:54 pm

### Re: "There is a rebalancing bonus ... false"

Rebalancing by selling or buying stocks is a risk story. Let's say that Mr. Boglehead inherits \$100K and decides that he's comfortable investing it by taking on an "average" level of investment risk as represented by the collective wisdom of the capital markets. For purposes of illustration, let's say that is currently represented by 60% equities and 40% bonds. Subsequently, there is a stock market crash in which stocks lose 50% of their value, while bonds gain 25% of their value. Now Mr. Boglehead is holding a portfolio worth \$80K and it is allocated approximately 40% in stocks and 60% in bonds. Mr. Boglehead decides to rebalance back to 60/40 by selling about \$20K in bonds to buy \$20K in stocks.

Mr. Boglehead has decided that he's smarter than the markets, which have revalued the relative values of stocks and bonds. For some reason, he believes that the market valuations of stocks and bonds before the market crash is closer to "truth" than the market valuations now. He's willing to increase the risk of his portfolio by buying stocks because they've lost value and sell bonds because they've gone up in value. Relative to his starting point, he's now invested \$80K in stocks and \$20K in bonds, and has an 80/20 allocation marked to his starting point. That is to say, if he'd invested his \$100K with an 80/20 allocation to begin with, he'd now have exactly the same allocation that he will have if he rebalances. Since he's increased the initial risk level of his portfolio, his expected return is higher -- but at the cost of taking more risk.
The fox knows many little things, but the hedgehog knows one big thing ~ Archilochus | | "To the man with just a hammer, every problem looks pretty much like a nail" ~ Charlie Munger

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: "There is a rebalancing bonus ... false"

I checked with my money. It does not remember where it came from.

Bogleheads 1 and 2 each get a windfall of \$100k.

Boglehead1 invests \$100k in the market at 60/40. Stocks crash by 50% and he rebalances his remaining \$70k back to 60/40.

Boglehead2 was on vacation when the check arrived, and so he did not invest until after the market crash. Nonetheless, he invests it all at 60/40 after the crash.

Please explain how Bogleheads 1 and 2 do not have the same level of risk with their 60/40 AA?

Keith
Déjà Vu is not a prediction

Browser
Posts: 4504
Joined: Wed Sep 05, 2012 4:54 pm

### Re: "There is a rebalancing bonus ... false"

Or, alternatively let's consider Mr. Boglehead who invests a \$100K windfall. He decides that 60% stocks and 40% bonds is an appropriate allocation strategy for him. Subsequently, the market crashes and stocks lose 50% while bonds gain 25%. He now has \$30K in stocks and \$50K in bonds = \$80K total. Luckily, he comes by another \$100K windfall. But instead of investing \$60K in stocks and \$40K in bonds, as he had previously done, he invests \$78K in stocks and \$22K in bonds -- a much riskier asset allocation than before. Why would he do that?
The fox knows many little things, but the hedgehog knows one big thing ~ Archilochus | | "To the man with just a hammer, every problem looks pretty much like a nail" ~ Charlie Munger

clevername
Posts: 257
Joined: Sun Jul 10, 2011 7:13 pm
Location: FL

### Re: "There is a rebalancing bonus ... false"

Browser wrote:Or, alternatively let's consider Mr. Boglehead who invests a \$100K windfall. He decides that 60% stocks and 40% bonds is an appropriate allocation strategy for him. Subsequently, the market crashes and stocks lose 50% while bonds gain 25%. He now has \$30K in stocks and \$50K in bonds = \$80K total. Luckily, he comes by another \$100K windfall. But instead of investing \$60K in stocks and \$40K in bonds, as he had previously done, he invests \$78K in stocks and \$22K in bonds -- a much riskier asset allocation than before. Why would he do that?

I was actually wondering the opposite. If the investor was initially happy with \$40,000 in bonds, why would the investor put a single penny into bonds from the new windfall? If the investor wanted more bonds s/he should have bought more with the first windfall. I don't understand why an AA has to be by percentage of portfolio instead of there being a threshold which, once attained, means the investor can stop buying cash/bonds/FI and just go for broke on stocks. The answer is probably to do with the investor's willingness to take risk (and manage emotions during a meltdown) rather than need and ability.

The investor in your example should be looking at the portfolio as a whole instead of each individual "pile" of assets. Investors actually do your dilemma all the time in the form of rebalancing with new contributions. The amount is smaller than a windfall, and the market rarely (never?) tumbles 50% overnight, but the example holds. Sometimes you need to buy all stocks with your deposit, sometimes all bonds, sometimes 50/50. How is this any different?

To keep this on topic: since stocks rarely (if ever) drop 50% overnight you should pull up excel and run some simulations about how it would actually go. Your stocks would drop 10%, then 5%, then 20%, then climb a bit, then drop another 15%, and so on. Set rebalancing bands at 5% or maybe 10% out of alignment, so ignore 64/46 but rebalance at 65/45. If the investor was responsible and true to the IPS, rebalancing whenever bands are hit, then it wouldn't have much effect. If the investor ignored the tanking equity portion, then "rebalanced" when stocks were 50% of their initial balance, then "rebalanced" again once stocks had recovered, then we've got a market timer and not a boglehead

umfundi
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### Re: "There is a rebalancing bonus ... false"

Personally, I think rebalancing is a tool to maintain your asset allocation. Which means, to manage and control your risk.

I think there is a persuasive argument that if stocks drop 50%, they are less risky than before.

Keith
Déjà Vu is not a prediction

Browser
Posts: 4504
Joined: Wed Sep 05, 2012 4:54 pm

### Re: "There is a rebalancing bonus ... false"

So then, to continue the saga let's do so with Unfundi's data: Investor A puts \$100K into 60% stocks, 40% bonds. The stock market tanks by 50% while bonds are steady. Investor A now has \$30K in stocks and \$40K in bonds, or a 42%/58% allocation. He rebalances his portfolio by selling \$12K in bonds to buy another \$12K in stocks. He now has a 60/40 allocation of his remaining \$70K. He has invested a total of \$72K in stocks over the course of the initial purchase plus rebalancing, or 72% of the initial \$100K. Investor B was unable to invest his \$100K until after the market crash. He invests using a 60/40 allocation, putting \$60K into stocks and \$40K into bonds. Over the same course of time he has put \$60K, or 60%, of his capital at risk in the stock market.

Only if you ignore the dollar-weighted amounts that are invested over time can you say that both Investor A and Investor B have taken equal investment risk. I'm not sure that anybody does that in the real world. Most of us consider the amount invested to be part of our risk profile. Surely, Bill Gates has more dollars at risk if he has 50% of \$67 billion invested in stocks than Joe Sixpack who has 50% of \$670 invested in stocks.
The fox knows many little things, but the hedgehog knows one big thing ~ Archilochus | | "To the man with just a hammer, every problem looks pretty much like a nail" ~ Charlie Munger

umfundi
Posts: 3361
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### Re: "There is a rebalancing bonus ... false"

Browser wrote:Or, alternatively let's consider Mr. Boglehead who invests a \$100K windfall. He decides that 60% stocks and 40% bonds is an appropriate allocation strategy for him. Subsequently, the market crashes and stocks lose 50% while bonds gain 25%. He now has \$30K in stocks and \$50K in bonds = \$80K total. Luckily, he comes by another \$100K windfall. But instead of investing \$60K in stocks and \$40K in bonds, as he had previously done, he invests \$78K in stocks and \$22K in bonds -- a much riskier asset allocation than before. Why would he do that?

Because his IPS says his AA should be 60/40, and what you describe restores that AA? Good Boglehead!

Keith
Déjà Vu is not a prediction

Browser
Posts: 4504
Joined: Wed Sep 05, 2012 4:54 pm

### Re: "There is a rebalancing bonus ... false"

Yes, but there is precious little evidence that restoring one's initial AA by rebalancing does anything much and it's not endorsed by Jack Bogle among others. It IS important to control one's portfolio risk level, so I agree with Bogle that if you find that the stock allocation has gotten way too high for you it might be a good idea to trim it back -- especially since the dollars at risk must have grown a lot too. If there is a "rebalancing bonus" I expect it can mostly be attributed to the fact that rebalancers are able to convince themselves to market time by buying more stocks because they've dropped. That increases portfolio risk, but it has paid off at least in the U.S. market because of mean reversion and the fact that over time stocks have gone up more than they've gone down. How well it has paid off for investors in Japanese stocks or other markets that have just laid there I don't know.
The fox knows many little things, but the hedgehog knows one big thing ~ Archilochus | | "To the man with just a hammer, every problem looks pretty much like a nail" ~ Charlie Munger

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: "There is a rebalancing bonus ... false"

Browser wrote:Yes, but there is precious little evidence that restoring one's initial AA by rebalancing does anything much and it's not endorsed by Jack Bogle among others. It IS important to control one's portfolio risk level, so I agree with Bogle that if you find that the stock allocation has gotten way too high for you it might be a good idea to trim it back -- especially since the dollars at risk must have grown a lot too. If there is a "rebalancing bonus" I expect it can mostly be attributed to the fact that rebalancers are able to convince themselves to market time by buying more stocks because they've dropped. That increases portfolio risk, but it has paid off at least in the U.S. market because of mean reversion and the fact that over time stocks have gone up more than they've gone down. How well it has paid off for investors in Japanese stocks or other markets that have just laid there I don't know.

Think what you may. I rebalanced through 2008-09 and came back to even in my total portfolio more than a year before the equity indexes reached their pre-crash levels.

Keith
Déjà Vu is not a prediction

YDNAL
Posts: 13773
Joined: Tue Apr 10, 2007 4:04 pm
Location: Biscayne Bay

### Re: "There is a rebalancing bonus ... false"

Browser wrote:Or, alternatively let's consider Mr. Boglehead who invests a \$100K windfall. He decides that 60% stocks and 40% bonds is an appropriate allocation strategy for him. Subsequently, the market crashes and stocks lose 50% while bonds gain 25%. He now has \$30K in stocks and \$50K in bonds = \$80K total. Luckily, he comes by another \$100K windfall. But instead of investing \$60K in stocks and \$40K in bonds, as he had previously done, he invests \$78K in stocks and \$22K in bonds -- a much riskier asset allocation than before. Why would he do that? (my emphasis)

That's not true (underlined).
• Original allocation 60/40, new allocation 60/40 Stock/Bond.
• New \$108/\$72K allocation -- after -50%/+25% market adjustments and new \$100K windfall -- has MUCH lower Stock Price/Earnings ratio than Original \$60/\$40K allocation. There is a slightly higher risk in Bond pricing.
Landy | Be yourself, everyone else is already taken -- Oscar Wilde

Posts: 6758
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### Re: "There is a rebalancing bonus ... false"

William J. Bernstein wrote:Is there any reason to believe that, on average, rebalancing will help more than it hurts? Not if we believe that market movements are random. After all, we rebalance with the hope that an asset with past higher/lower than average returns will have future lower/higher than average returns.

1. Bernstein is wrong when his says that there is no rebalancing bonus in a random market.

Shannon's coin-flip-bet example (cited earlier) shows a rebalancing bonus in a random market:

http://www.stableinvesting.com/2013/04/ ... demon.html

If you don't rebalance, your expected CAGR or geometric return is zero. If you rebalance, it's positive, it's around 5% per bet.

2. Bernstein is also wrong that we have to hope for future higher/lower returns for rebalancing to work. Shannon's example shows that it works even if there is no expectation of higher or lower returns. The expected return on each coin flip bet is 25% whether you rebalance or not, and yet you get a bonus for rebalancing. In the Shannon example, rebalancing gives a bonus simply because it's a superior money management strategy for exploiting a favorable bet (as in the Kelly Criterion) relative to not rebalancing.

The paper linked at the end of the stableinvesting blog, gives an analysis closer to the real stock market:

http://www.iinews.com/site/pdfs/JWM_Fal ... metric.pdf

Of course, Bernstein is blundering on what are merely theoretical points, and I am merely giving the rebuttal of these theoretical points. This does not prove that one should expect a rebalancing bonus when investing your nest egg in the stock market.

camontgo
Posts: 316
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Location: Chicago, IL
Contact:

### Re: "There is a rebalancing bonus ... false"

William J. Bernstein wrote:Is there any reason to believe that, on average, rebalancing will help more than it hurts? Not if we believe that market movements are random. After all, we rebalance with the hope that an asset with past higher/lower than average returns will have future lower/higher than average returns.

1. Bernstein is wrong when his says that there is no rebalancing bonus in a random market.

In an earlier discussion on rebalancing, I posted simulation results on rebalancing vs. not-rebalancing for two assets which follow a random walk (normally distributed annual returns). The simulation is pretty much identical to one of the simulations on Bernstein's website...I recreated his simulation so I could try with return correlation (posted on my blog) and other parameters he didn't simulate.

I found that, over a given horizon, there is a rebalancing bonus if the realized returns end up being similar for two uncorrelated assets with the same expected returns. This happens because the geometric mean is higher for the rebalanced portfolio as several have noted above.

However, with a random walk, the realized returns of the two portfolios often diverge significantly over a given horizon. In my simulation, I show that over ten years two uncorrelated assets with 10% expected return and 20% standard deviation can diverge significantly when it comes to realized returns....and if they do there is likely to be a rebalancing penalty. In simulation, the average rebalancing "bonus" over 1000 trials was zero.

Over longer horizons, the result remains the same. The likelihood of the average returns being close increases with a longer horizon..but the "realized return difference" window where you get a bonus shrinks....so the average bonus over many trials is still zero.

Where have I gone wrong with this analysis? I understand that you can get a different result when the return distribution is not normal (i.e. the Shannon example)....and there is often a bonus in practice when two very volatile assets end up having average returns that are in the same ballpark...but with a random walk model two very volatile assets can also end up having hugely different returns, so I don't think there is an expected bonus if the returns follow a random walk.
Last edited by camontgo on Fri May 31, 2013 2:41 pm, edited 1 time in total.
"Essentially, all models are wrong, but some are useful." - George E. P Box

umfundi
Posts: 3361
Joined: Tue Jun 07, 2011 5:26 pm

### Re: "There is a rebalancing bonus ... false"

camontgo wrote:
William J. Bernstein wrote:Is there any reason to believe that, on average, rebalancing will help more than it hurts? Not if we believe that market movements are random. After all, we rebalance with the hope that an asset with past higher/lower than average returns will have future lower/higher than average returns.

1. Bernstein is wrong when his says that there is no rebalancing bonus in a random market.

In an earlier discussion on rebalancing, I posted simulation results on rebalancing vs. not-rebalancing for two assets which follow a random walk (normally distributed annual returns). The simulation is pretty much identical to one of the simulations on Bernstein's website...I recreated his simulation so I could try with return correlation (posted on my blog) and other parameters he didn't simulate.

I found that, over a given horizon, there is a rebalancing bonus if the realized return of two uncorrelated assets with the same expected return end up having a similar realized return. This happens because the geometric mean is higher for the rebalanced portfolio as several have noted above.

However, with a random walk, the realized returns of the two portfolios often diverge significantly over a given horizon. In my simulation, I show that over ten years two uncorrelated assets with 10% expected return and 20% standard deviation can diverge significantly when it comes to realized returns....and if they do there is likely to be a rebalancing penalty. In simulation, the average rebalancing "bonus" over 1000 trials was zero.

Over longer horizons, the result remains the same. The likelihood of the average returns being close increases with a longer horizon..but the "realized return difference" window where you get a bonus shrinks....so the average bonus over many trials is still zero.

Where have I gone wrong with this analysis? I understand that you can get a different result when the return distribution is not normal (i.e. the Shannon example)....and there is often a bonus in practice when two very volatile assets end up having average returns that are in the same ballpark...but with a random walk model two very volatile assets can also end up having hugely different returns, so I don't think there is an expected bonus if the returns follow a random walk.

I, for one, think you have it exactly right.

Keith
Déjà Vu is not a prediction

swaption
Posts: 1084
Joined: Tue Jul 29, 2008 11:48 am

### Re: "There is a rebalancing bonus ... false"

William J. Bernstein wrote:Is there any reason to believe that, on average, rebalancing will help more than it hurts? Not if we believe that market movements are random. After all, we rebalance with the hope that an asset with past higher/lower than average returns will have future lower/higher than average returns.

1. Bernstein is wrong when his says that there is no rebalancing bonus in a random market.

Shannon's coin-flip-bet example (cited earlier) shows a rebalancing bonus in a random market:

http://www.stableinvesting.com/2013/04/ ... demon.html

If you don't rebalance, your expected CAGR or geometric return is zero. If you rebalance, it's positive, it's around 5% per bet.

2. Bernstein is also wrong that we have to hope for future higher/lower returns for rebalancing to work. Shannon's example shows that it works even if there is no expectation of higher or lower returns. The expected return on each coin flip bet is 25% whether you rebalance or not, and yet you get a bonus for rebalancing. In the Shannon example, rebalancing gives a bonus simply because it's a superior money management strategy for exploiting a favorable bet (as in the Kelly Criterion) relative to not rebalancing.

The paper linked at the end of the stableinvesting blog, gives an analysis closer to the real stock market:

http://www.iinews.com/site/pdfs/JWM_Fal ... metric.pdf

Of course, Bernstein is blundering on what are merely theoretical points, and I am merely giving the rebuttal of these theoretical points. This does not prove that one should expect a rebalancing bonus when investing your nest egg in the stock market.

Thanks, the II paper is very good, and obviously I agree with the conclusions. But some things continue to bother me in all this. First of all, I think the paper is weak when it comes to their addressing who is on the other side of the trade. They take a whole behavioral kind of arbitrage route that does not come across as compelling. But the whole concept of geometric returns is what really bothers me. Take the example of Starbucks and Apple. Is there some bias as a result of this selection? What if one of the companies had gone bankrupt during this time period leaving the stock with a value of 0. One would have essentially been rebalancing into oblivion. That can't happen in a geometric model of returns.

I say the above because I always come back to another concept I embrace, the notion that money invested in equities is just plain at risk. Perhaps we can easily dismiss the idea of the entire equity market being worthless. But the notion that one could literally rebalance into oblivion just gives me pause when I think of things being characterized in geometric terms. Put another way, is the empitical evidence merely a by product of dealing with a universe of outcomes that is not comprehensive enough to include some really bad results? Once again consider the example of Apple and Starbucks, which they seem to think is beneficial as a model, but as I think I pointed out really is not appropriate.

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### Re: "There is a rebalancing bonus ... false"

camontgo wrote:
William J. Bernstein wrote:Is there any reason to believe that, on average, rebalancing will help more than it hurts? Not if we believe that market movements are random. After all, we rebalance with the hope that an asset with past higher/lower than average returns will have future lower/higher than average returns.

1. Bernstein is wrong when his says that there is no rebalancing bonus in a random market.

In an earlier discussion on rebalancing, I posted simulation results on rebalancing vs. not-rebalancing for two assets which follow a random walk (normally distributed annual returns). The simulation is pretty much identical to one of the simulations on Bernstein's website...I recreated his simulation so I could try with return correlation (posted on my blog) and other parameters he didn't simulate.

I found that, over a given horizon, there is a rebalancing bonus if the realized return of two uncorrelated assets with the same expected return end up having a similar realized return. This happens because the geometric mean is higher for the rebalanced portfolio as several have noted above.

However, with a random walk, the realized returns of the two portfolios often diverge significantly over a given horizon. In my simulation, I show that over ten years two uncorrelated assets with 10% expected return and 20% standard deviation can diverge significantly when it comes to realized returns....and if they do there is likely to be a rebalancing penalty. In simulation, the average rebalancing "bonus" over 1000 trials was zero.

Over longer horizons, the result remains the same. The likelihood of the average returns being close increases with a longer horizon..but the "realized return difference" window where you get a bonus shrinks....so the average bonus over many trials is still zero.

Where have I gone wrong with this analysis? I understand that you can get a different result when the return distribution is not normal (i.e. the Shannon example)....and there is often a bonus in practice when two very volatile assets end up having average returns that are in the same ballpark...but with a random walk model two very volatile assets can also end up having hugely different returns, so I don't think there is an expected bonus if the returns follow a random walk.

You have gone wrong with the statement "the average bonus over many trials is zero". You likely did not get exactly zero as the estimate from your simulations. And, in any case, you certainly did not prove that the average bonus was zero because it's not possible to do that with a simulation. You could design a simulation that rejects a hypothesis like "The average bonus is greater than N%" with some specific level of certainty, but that does not prove it's zero with any level of certainty. You might find that a sufficiently large simulation would tighten the uncertainty on the average bonus around a non-zero value sufficient to prove it was not zero.

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### Re: "There is a rebalancing bonus ... false"

swaption wrote:Thanks, the II paper is very good, and obviously I agree with the conclusions. But some things continue to bother me in all this. First of all, I think the paper is weak when it comes to their addressing who is on the other side of the trade. They take a whole behavioral kind of arbitrage route that does not come across as compelling. But the whole concept of geometric returns is what really bothers me. Take the example of Starbucks and Apple. Is there some bias as a result of this selection? What if one of the companies had gone bankrupt during this time period leaving the stock with a value of 0. One would have essentially been rebalancing into oblivion. That can't happen in a geometric model of returns.

I say the above because I always come back to another concept I embrace, the notion that money invested in equities is just plain at risk. Perhaps we can easily dismiss the idea of the entire equity market being worthless. But the notion that one could literally rebalance into oblivion just gives me pause when I think of things being characterized in geometric terms. Put another way, is the empitical evidence merely a by product of dealing with a universe of outcomes that is not comprehensive enough to include some really bad results? Once again consider the example of Apple and Starbucks, which they seem to think is beneficial as a model, but as I think I pointed out really is not appropriate.

I think the possibility of a stock going to zero and staying there is not really a problem with geometric returns. It's a problem for other required assumptions: the assumptions about the returns, variance, and covariance of the assets.

camontgo
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### Re: "There is a rebalancing bonus ... false"

tadamsmar wrote:You have gone wrong with the statement "the average bonus over many trials is zero". You likely did not get exactly zero as the estimate from your simulations. And, in any case, you certainly did not prove that the average bonus was zero because it's not possible to do that with a simulation. You could design a simulation that rejects a hypothesis like "The average bonus is greater than N%" with some specific level of certainty, but that does not prove it's zero with any level of certainty. You might find that a sufficiently large simulation would tighten the uncertainty on the average bonus around a non-zero value sufficient to prove it was not zero.

I agree that my simulation doesn't amount to "proof" and the average for 1000 trials is not precisely zero....so I'd reword the statement to "very close to zero".

To be more specific, I just ran 1000 trials, and the average rebalancing bonus over 10 years for two assets with 10% return and 20% standard deviation was 0.05% per year. The simulation does show a rebalancing bonus about 2/3 of the time...and a penalty only about 1/3 of the time...but the penalties tend to be larger than the bonuses.
"Essentially, all models are wrong, but some are useful." - George E. P Box