"There is a rebalancing bonus ... false"
"There is a rebalancing bonus ... false"
This article seems to rail against the notion that there is a bonus by rebalancing.
http://www.retailinvestor.org/why/bonus.html
Similarly, this post proves mathematically that via continuous rebalancing, there is no bonus from "buying low and selling high", as the final value of a portfolio depends on only its proportional returns.
viewtopic.php?t=63015
Rebalancing to your target asset allocation to maintain constant risk exposure is completely valid. However, this is separate from the idea of a "bonus."
I have seen arguments that rebalancing does have a bonus if the market exhibits momentum, but I'm not sure how long a momentumbased strategy will last in the age of highfrequency trading. For those that do believe there is a rebalancing bonus, how do you justify it?
http://www.retailinvestor.org/why/bonus.html
Similarly, this post proves mathematically that via continuous rebalancing, there is no bonus from "buying low and selling high", as the final value of a portfolio depends on only its proportional returns.
viewtopic.php?t=63015
Rebalancing to your target asset allocation to maintain constant risk exposure is completely valid. However, this is separate from the idea of a "bonus."
I have seen arguments that rebalancing does have a bonus if the market exhibits momentum, but I'm not sure how long a momentumbased strategy will last in the age of highfrequency trading. For those that do believe there is a rebalancing bonus, how do you justify it?

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Re: "THERE IS A REBALANCING BONUS ... FALSE"
In total over the 51 years there has been a 0.37% annualized benefit  assuming zero costs for rebalancing.
Well, I guess I can get well more than 2x my annual ER by rebalancing. That sounds like a bonus to me?
Also, this article doesn't seem to get into rebalancing via bands, which I do, at any rate.
And..I love this link at the bottom of the page: http://www.retailinvestor.org/why/timing.html
That page needs more than a few grains of salt.
Re: "There is a rebalancing bonus ... FALSE"
boggler wrote:For those that do believe there is a rebalancing bonus, how do you justify it?
Aren't you purposely restricting your captive audience ?
Landy  Be yourself, everyone else is already taken  Oscar Wilde
Re: "There is a rebalancing bonus ... FALSE"
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.
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.
Re: "There is a rebalancing bonus ... FALSE"
From the first link
We've been over this one before. Managing risk does not mean reducing risk, it means managing risk. You don't want your portfolio to get too risky but you don't want it to get too safe either.
Similarly, improving your riskadjusted return may not be the same as maximising your real return.
But what happens when your 'safe' asset class has outperformed, and now overpowers your portfolio? Rebalancing will not reduce risk because you are INCREASING the risky asset class.
We've been over this one before. Managing risk does not mean reducing risk, it means managing risk. You don't want your portfolio to get too risky but you don't want it to get too safe either.
Similarly, improving your riskadjusted return may not be the same as maximising your real return.
Re: "There is a rebalancing bonus ... FALSE"
Poorly written article, essentially data mining a case against what it refers to as data mining. I refer to the following Vanguard paper referenced in the article:
http://www.retailinvestor.org/pdf/vanguard.pdf
The relevant data is in Table 3. This compares a 60/40 portfolio rebalanced and not rebalanced from 1960  2003. There is a bunch of stuff about rebalance frequency, but I'll leave it at annually. You are correct, the not rebalnced portfolio returned an average of 4 bps more per annum (i.e 0.04%). But the average equity allocation of such a portfolio is 74% as compared to 61% for the rebalanced portfolio, with a volatility that is roughly 20% higher (12% vs 10%). To me, this is a bonus. So I'm not saying if it exists or not, but I do find it somewhat amusing to read someone making such a strong case using data that absolutely does not support their position..
http://www.retailinvestor.org/pdf/vanguard.pdf
The relevant data is in Table 3. This compares a 60/40 portfolio rebalanced and not rebalanced from 1960  2003. There is a bunch of stuff about rebalance frequency, but I'll leave it at annually. You are correct, the not rebalnced portfolio returned an average of 4 bps more per annum (i.e 0.04%). But the average equity allocation of such a portfolio is 74% as compared to 61% for the rebalanced portfolio, with a volatility that is roughly 20% higher (12% vs 10%). To me, this is a bonus. So I'm not saying if it exists or not, but I do find it somewhat amusing to read someone making such a strong case using data that absolutely does not support their position..
Re: "There is a rebalancing bonus ... FALSE"
Here is the definitive (IMO) article on the rebalancing bonus, with math, etc
http://www.efficientfrontier.com/ef/996/rebal.htm
http://www.efficientfrontier.com/ef/996/rebal.htm
Re: "There is a rebalancing bonus ... FALSE"
I would argue that over a 30 or 40 year time horizon, 0.37% is quite significant.

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Re: "There is a rebalancing bonus ... FALSE"
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.
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.
"The stock market [fluctuation], therefore, is noise. A giant distraction from the business of investing.”  Jack Bogle
 Rick Ferri
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Re: "There is a rebalancing bonus ... FALSE"
Rebalancing narrows the range of portfolio risk. The article does not address this benefit.
Rick Ferri
Rick Ferri
Choose a few lowcost index funds in different asset classes, rebalance occasionally and forgetaboutit!

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Re: "There is a rebalancing bonus ... FALSE"
staythecourse wrote:The fact remains is OVER time there is NO way a riskier asset class will underperform a safe asset class.
One quibble, but that depends on the start and end points. Since we do not have unlimited time there has to be a balance between risk and reward. Higher risk should yield higher return but it doesn't always do so within the finite boundaries of our lives.

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Re: "There is a rebalancing bonus ... FALSE"
The "article" starts with a false premise: rebalancing is promoted as an unfailing technique for improving returns (i.e., bonus). Beating up a straw man. Any time you keep a fixed percent in a safer class with lower longterm returns your very longterm returns will be lower than if you allowed the position to deteriorate over a very long span of time. If my timeline was 70 years, I'd go 100% stocks. Mine is 20an entirely different ball game.
Glancing over some of the other offerings makes me wonder what the motive of that site is. Satire?
The irony was not lost on me, MindBogler
Glancing over some of the other offerings makes me wonder what the motive of that site is. Satire?
The irony was not lost on me, MindBogler
Don't do something. Just stand there!
Re: "There is a rebalancing bonus ... FALSE"
I use:
50% domestic SCV
25% international developed exUS SCV
25% EMSCV
The more volatility you add, the larger the bonus. It also helps if all the assets have similar long term expected returns. When I backtest this mix 6 ways to Sunday, the results are consistently around a rebalancing bonus of ~0.75%.
50% domestic SCV
25% international developed exUS SCV
25% EMSCV
The more volatility you add, the larger the bonus. It also helps if all the assets have similar long term expected returns. When I backtest this mix 6 ways to Sunday, the results are consistently around a rebalancing bonus of ~0.75%.
There are no guarantees, only probabilities.
 nisiprius
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Re: "There is a rebalancing bonus ... FALSE"
I think wbern has been fairly clear about this. The article cited by the original poster mentioned Bernstein, and I think that if he didn't originate the phrase "rebalancing bonus," he played a big part in publicizing it. There's a rebalancing bonus if the assets in question exhibit mean reversion over the time interval during which rebalancing is done. There isn't if there isn't. And there is a "rebalancing penalty" if they exhibit momentum over that time period. It all reduces to the question of whether asset classes really truly do exhibit mean reversion. His words:
It's a surprisingly subtle effect, though. It's not magic that ratchets up your return by a factor of two by automatically buying stocks in 20082009.
It's not so clear to me that there's always a risk reduction effect. During 20082009 I was annoyed that during the fall, my portfolio was consistently falling faster than the fall in the S&P multiplied by my stock allocation, and I traced it to rebalancing in my Vanguard Balanced Index fund. During a sustained drop, rebalancing amplifies the effect of the drop by constantly throwing good money after bad; if it is followed by a rise (mean reversion!) it amplifies the rise, as assets bought low are constantly being automatically sold off higher.
The people that do lots of backtesting and announce optimum strategies for rebalancing do seem to me to resemble market timers endlessly tweaking their moving average systems...
"Is this actually true? Probably."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.
Is this actually true? Probably. Recall that over short periods of time asset classes display momentum, but that over periods of time over a year or longer tend to meanrevert....
Rebalance your portfolio approximately once every few years; more than once per year is probably too often. In taxable portfolios, do so even less frequently.
It's a surprisingly subtle effect, though. It's not magic that ratchets up your return by a factor of two by automatically buying stocks in 20082009.
It's not so clear to me that there's always a risk reduction effect. During 20082009 I was annoyed that during the fall, my portfolio was consistently falling faster than the fall in the S&P multiplied by my stock allocation, and I traced it to rebalancing in my Vanguard Balanced Index fund. During a sustained drop, rebalancing amplifies the effect of the drop by constantly throwing good money after bad; if it is followed by a rise (mean reversion!) it amplifies the rise, as assets bought low are constantly being automatically sold off higher.
The people that do lots of backtesting and announce optimum strategies for rebalancing do seem to me to resemble market timers endlessly tweaking their moving average systems...
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: "There is a rebalancing bonus ... FALSE"
What's the net effect on funds offered by Vanguard, TRP and others that maintain a fixed allocation to stocks and bonds?
Larry Swedroe has written on this site a couple of times that in the absence of "frictions" (taxes, costs) that rebalancing daily would be the optimal strategy.
I suppose the case can be made that from a "risk management" perspective that may be the case, but not from a momentum perspective (wherein rebalancing bands might be more optimal).
Since it's sorta hard to know the future it's difficult to decide which is the best strategy at any given point in time.
Larry Swedroe has written on this site a couple of times that in the absence of "frictions" (taxes, costs) that rebalancing daily would be the optimal strategy.
I suppose the case can be made that from a "risk management" perspective that may be the case, but not from a momentum perspective (wherein rebalancing bands might be more optimal).
Since it's sorta hard to know the future it's difficult to decide which is the best strategy at any given point in time.
“Tactics without strategy is the noise before defeat.”  Sun Tzu  "Everybody has a plan until they get punched in the mouth."  Mike Tyson
Re: "There is a rebalancing bonus ... FALSE"
nisi,
Look at the link that I posted earlier. There is no mean reversion in that example but there is a positive effect associated with rebalancing. It's an issue of arithmetic versus geometric returns. Mean reversion can certainly help, but it is not a necessary condition for rebalancing to boost returns.
Look at the link that I posted earlier. There is no mean reversion in that example but there is a positive effect associated with rebalancing. It's an issue of arithmetic versus geometric returns. Mean reversion can certainly help, but it is not a necessary condition for rebalancing to boost returns.
 nisiprius
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Re: "There is a rebalancing bonus ... FALSE"
I don't have time to look at it now, the arithmetic/geometric thing makes my head spin and it always takes an hour to clear it, but I'm almost sure you're wrong. Every time I've looked at it before, any apparent rebalancing bonus from random series always turned out involve a smuggledin mean reversion assumption.rmelvey wrote:nisi,
Look at the link that I posted earlier. There is no mean reversion in that example but there is a positive effect associated with rebalancing. It's an issue of arithmetic versus geometric returns. Mean reversion can certainly help, but it is not a necessary condition for rebalancing to boost returns.
[Added} I think I see where it is. You have to consider, not just the effect when "there are an equal number of heads and tails," and the catch is in the tendency to make the unconscious assumption that since there tend to be roughly equal numbers of heads and tails, the results must be roughly similar to what happens when there are equal numbers. Monte Carlo simulations won't cut it because it's a Martingalelike situation: you have to be sure that you have included the properly weighted effect of the the very rare sequences that are almost all heads or almost all tails. And of course if there is mean reversion, then you get overweighting of the number of runs where heads and tails are roughly equal so you do get a bonus.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
 nisiprius
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Re: "There is a rebalancing bonus ... FALSE"
Well, in fact I tried that experiment in real life and it didn't work. That is to say, I derived no advantage from having been in Balanced Index through 20082009 versus being in unrebalanced separate holdings of Total Stock and Total Bond. If you look at growth chart for Balanced Index Fund and select a day before and a day after 20082009 in which the starting and points are as close to equal as possible, and then calculate the final value for an unrebalanced mix that started as 60% Total Stock, 40% Total Bond, the difference (on a $10,000 investment) is less than $60.Blues wrote:Larry Swedroe has written on this site a couple of times that in the absence of "frictions" (taxes, costs) that rebalancing daily would be the optimal strategy....
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
 neurosphere
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Re: "There is a rebalancing bonus ... false"
What if one rebalances between two asset classes with similar risk/return profiles? For example, one could hold an international developed markets fund, or instead hold a european and pacific fund and rebalance.
Suppose the future returns are the same between two asset classes. In that case shouldn't rebalancing increase returns assuming that the assets don't move in lockstep from point A to point B? I thought THAT was the point of the rebalancing bonus? To capture uncorrelated movements between two asset classes with similar returns.
NS
Suppose the future returns are the same between two asset classes. In that case shouldn't rebalancing increase returns assuming that the assets don't move in lockstep from point A to point B? I thought THAT was the point of the rebalancing bonus? To capture uncorrelated movements between two asset classes with similar returns.
NS
 Sotirios Keros. "I don't always buy investments, but when I do, I prefer them to be boring. Sleep well, my friends." The Least Interesting Investor in the World
Re: "There is a rebalancing bonus ... false"
This is the best analysis I've seen of the rebalancing bonus.
http://www.iinews.com/site/pdfs/JWM_Fal ... metric.pdf
Basically, the "bonus" relies upon the assumption that the markets are meanreverting. If they are, then buying low and selling high works, otherwise, it doesn't.
http://www.iinews.com/site/pdfs/JWM_Fal ... metric.pdf
Overall, diversifying and rebalancing is a valu able discipline and can be used to exploit volatility. Theoretically, rebalancing reduces concentration risk, downside risk, and volatility, while increasing the long term growth rate of the portfolio. In practice, it creates a contrarian trading pattern that trades against natural investor tendencies and takes advantage of volatility, reversals, and other return characteristics.
Basically, the "bonus" relies upon the assumption that the markets are meanreverting. If they are, then buying low and selling high works, otherwise, it doesn't.
Re: "There is a rebalancing bonus ... FALSE"
nisiprius wrote:Well, in fact I tried that experiment in real life and it didn't work. That is to say, I derived no advantage from having been in Balanced Index through 20082009 versus being in unrebalanced separate holdings of Total Stock and Total Bond. If you look at growth chart for Balanced Index Fund and select a day before and a day after 20082009 in which the starting and points are as close to equal as possible, and then calculate the final value for an unrebalanced mix that started as 60% Total Stock, 40% Total Bond, the difference (on a $10,000 investment) is less than $60.Blues wrote:Larry Swedroe has written on this site a couple of times that in the absence of "frictions" (taxes, costs) that rebalancing daily would be the optimal strategy....
Interesting, thanks for sharing that data. Seems like a "no blood, no foul" outcome in your case.
I wonder what the long term results will be for a target allocation fund when it rebalances TSM, TI, TBM and the upcoming Int'l Bond fund on a daily basis going forward.
“Tactics without strategy is the noise before defeat.”  Sun Tzu  "Everybody has a plan until they get punched in the mouth."  Mike Tyson
Re: "There is a rebalancing bonus ... false"
boggler wrote:This is the best analysis I've seen of the rebalancing bonus.
http://www.iinews.com/site/pdfs/JWM_Fal ... metric.pdfOverall, diversifying and rebalancing is a valu able discipline and can be used to exploit volatility. Theoretically, rebalancing reduces concentration risk, downside risk, and volatility, while increasing the long term growth rate of the portfolio. In practice, it creates a contrarian trading pattern that trades against natural investor tendencies and takes advantage of volatility, reversals, and other return characteristics.
Basically, the "bonus" relies upon the assumption that the markets are meanreverting. If they are, then buying low and selling high works, otherwise, it doesn't.
I don't understand how the coin flip example is "mean reverting" though. Each event is totally independent of prior moves in that experiment.
The intuition is that volatility creates a drag on average arithmetic returns, lowering the realized geometric average returns. Diversification and rebalancing lowers volatility, reducing the drag, therefore increasing the realized geometric average returns.
How come every time I try to explain this on the forum everyone treats me like a crazy person
Re: "There is a rebalancing bonus ... false"
rmelvey wrote:boggler wrote:This is the best analysis I've seen of the rebalancing bonus.
http://www.iinews.com/site/pdfs/JWM_Fal ... metric.pdfOverall, diversifying and rebalancing is a valu able discipline and can be used to exploit volatility. Theoretically, rebalancing reduces concentration risk, downside risk, and volatility, while increasing the long term growth rate of the portfolio. In practice, it creates a contrarian trading pattern that trades against natural investor tendencies and takes advantage of volatility, reversals, and other return characteristics.
Basically, the "bonus" relies upon the assumption that the markets are meanreverting. If they are, then buying low and selling high works, otherwise, it doesn't.
I don't understand how the coin flip example is "mean reverting" though. Each event is totally independent of prior moves in that experiment.
The intuition is that volatility creates a drag on average arithmetic returns, lowering the realized geometric average returns. Diversification and rebalancing lowers volatility, reducing the drag, therefore increasing the realized geometric average returns.
How come every time I try to explain this on the forum everyone treats me like a crazy person
Curious to know what you think of this:
viewtopic.php?p=351711#p351613
 Clearly_Irrational
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Re: "There is a rebalancing bonus ... false"
When I modeled it using monte carlo analysis rebalancing actually reduced returns slightly, however it reduced volatility at about a 3:1 ratio for the amount of return you traded away. On a risk adjusted basis rebalancing came out better.
 Epsilon Delta
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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.
Arithmetic and geometric means of what? All I see from any of the coin flip methods is a final distribution with a mean return of zero.
Re: "There is a rebalancing bonus ... false"
The coin flip game has an average arithmetic return of 25%
 Epsilon Delta
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Re: "There is a rebalancing bonus ... false"
rmelvey wrote:The coin flip game has an average arithmetic return of 25%
Let me repeat myself. What are you averaging? And I will probably add why should I care about that particular statistic? The final distribution always has zero expected return, your just rearranging the tails.
Re: "There is a rebalancing bonus ... false"
Epsilon Delta wrote:rmelvey wrote:The coin flip game has an average arithmetic return of 25%
Let me repeat myself. What are you averaging? And I will probably add why should I care about that particular statistic? The final distribution always has zero expected return, your just rearranging the tails.
Hey ED hopefully this post will clear it all up for you
Okay let's look at the game again. You have a 0.5 probability of doubling your money (100% gain) and a 0.5 probability of losing half of your money (50% gain). What is the expected return of this game? It is the the sum of the probabilities multiplied by the corresponding % moves. So it is (0.5)*(100%)+(0.5)*(50%)=25%.
Let's look at using dollars, so you can clearly see that the game has potential to be a winner. If you have $100 someone is offering a 50/50 chance of getting another $100 or losing $50. Clearly this game has potential for winning, but not if you gamble all of your bankroll each round.
25% is your expected gain from playing the game for a single period. However, the volatility associated with the returns means that it doesn't compound in a favorable way if repeated over mutliple periods reinvesting all of your money. So the arithmetic average return of the game is 25%, but the geometric average return is 0%.
By choosing to rebalance between half cash and half coin flip, you cut the devations of the returns in half. The expected arithmetic return of the game becomes 12.5%, and the geometric return becomes positive.
Expected Average Arithmetic Return (100% coin flip): (1.5)/2 = 25%
Expected Geometric Return (100% coin flip): ((1 + 1)*(1+ .5))^.51 = 0
Expected Average Arithmetic Return (50% cash 50% coin flip): (.5.25)/2 = 12.5%
Expected Geometric Return (50% cash 50% coin flip): ((1 + 0.5)*(1+ .25))^.51 = 6.06%
Hopefully that clears it up
Can I get a CFA in here to back me up please? It's not like I am making this stuff up

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Re: "There is a rebalancing bonus ... false"
One Boglehead asked me to comment on this
Sorry but don't have time to read the whole thread
But we have discussed this many times
1) rebalancing doesn't increase RETURNS, unless you are rebalancing assets with the same expected returns and the returns in fact turn out to be at least similar.
2) what rebalancing does is produce a diversification return, meaning that the portfolio's return will be greater than the weighted average return of it' component parts.
The best example of that is in commoditiessee this post http://www.cbsnews.com/8301505123_16257585316/howcommoditiescanhelpaportfolio/
3)The role of rebalancing is to control risk, not increase returns
Larry
Sorry but don't have time to read the whole thread
But we have discussed this many times
1) rebalancing doesn't increase RETURNS, unless you are rebalancing assets with the same expected returns and the returns in fact turn out to be at least similar.
2) what rebalancing does is produce a diversification return, meaning that the portfolio's return will be greater than the weighted average return of it' component parts.
The best example of that is in commoditiessee this post http://www.cbsnews.com/8301505123_16257585316/howcommoditiescanhelpaportfolio/
3)The role of rebalancing is to control risk, not increase returns
Larry
Re: "There is a rebalancing bonus ... false"
The issue is risk, not making a few bucks here and there. Some folks 70 years of age who never rebalanced and find themselves 80% in stocks, well, you get the picture. I don't figure to make a buck rebalancing. I DO figure to preserve wealth by doing so.
 nisiprius
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Re: "There is a rebalancing bonus ... false"
rmelvey: I've figured out to my satisfaction, but probably not yours, why your "Shannon's Demon" example is misleading.
Let's consider one flip.
$100; bet $50; H $150, T $75.
Average: $112.50.
So the expectation of the game is a 12.5% return per flip.
Let's consider two flips.
H, $150, bet, $75; H, $225, T, $112.50.
T, $75; bet, $37.50; H, $112.50, T, 56.25.
Average: $126.5625.
Well, that's just what we expect: 100 + 1.125 * 1.125 = $126.5625. No rebalancing bonus, just the intrinsic 12.5% per flip.
Let's consider three flips.
HH, $225, bet, $112.50. H, $337.50; T, $168.75.
HT or TH, $112.50; bet, $56.75; H, 168.75; T, $84.75.
TT, $56.25; bet, $28.125; H, $84.75; T, $42.1825;
Average: $142.382813.
Well, that's just what we expect: 100 +1.125 * 1.125 * 1.125 = $142.382813. No rebalancing bonus, just the intrinsic 12.5% per flip.
Here's where I think the mental problem comes: "For a player who gambles his entire bankroll each round, it appears to be a wash. No matter the order of returns, if there are an equal number of heads and tails, the player ends up having exactly as much as he did at the start." That makes it sound as if it is a wash for the player who gambles his entire bankroll each round if the coin flips are random, "since" by the law of averages there will be equal numbers of heads and tails.
That's completely wrong. The game in which the player bets his whole stake is not a wash; it is hugely favorable to the player. His expectation is an equal chance of having either $200 or $50, average $125. His mathematical expectation is to increase his holdings by 25% per flip. That would mean that the player who stakes all should do better than the player who stakes half, and this is true. Let's run through it again, including all combinations, not just those with equal numbers of heads and tails.
One flip:
$100 > H, $200, T, $50
Average, $125.
Two flips:
H, $200; H, $400, T, $100.
T, $50. H, $100, T, $25.
Average: $156.50.
The equal headsandtails runs left him back where he started, but because he won more on HH than he lost on TT, he made money overall. Indeed, he made $100 * 1.25 * 1.25 = $156.50.
For three flips, it works out to $195.625 but I'm not going to show that one.
I'm going to go out to four flips to get some of the equal head/tail runs.
HHHH $1,600.00
HHHT $400.00
HHTH $400.00
HTHH $400.00
THHH $400.00
HHTT $100.00
HTHT $100.00
HTTH $100.00
THHT $100.00
THTH $100.00
TTHH $100.00
HTTT $25.00
THTT $25.00
TTHT $25.00
TTTH $25.00
TTTT $6.25
Average: $244.14
Again, just what's expected: 100 * 1.25^4 = $244.14.
The boldfaced entries are those with equal numbers of heads and tails, and those are a wash for the player, but the game is hugely in his favor overall because the runs with more heads far more than outweigh the runs with more tails.
The hidden "reversion to mean" assumption comes from showing examples in which there were equal numbers of heads and tails. If you're guaranteed equal numbers of heads and tails, then betting half and rebalancing beats betting all each time. If, however, heads and tails are in fact random, then when you bet it all you earn 25% per flip on the average, and when you bet half you earn 12.5% per flip on the average.
The false logic is this:
a) If there are equal numbers of heads and tails, it's a wash for the player who bets it all, but the player who bets half comes out ahead and thus does better than the player who bets it all.
b) On the average, there will be equal numbers of heads and tails.
c) Therefore on the average the player who bets half will do better than the player who bets it all.
Let's consider one flip.
$100; bet $50; H $150, T $75.
Average: $112.50.
So the expectation of the game is a 12.5% return per flip.
Let's consider two flips.
H, $150, bet, $75; H, $225, T, $112.50.
T, $75; bet, $37.50; H, $112.50, T, 56.25.
Average: $126.5625.
Well, that's just what we expect: 100 + 1.125 * 1.125 = $126.5625. No rebalancing bonus, just the intrinsic 12.5% per flip.
Let's consider three flips.
HH, $225, bet, $112.50. H, $337.50; T, $168.75.
HT or TH, $112.50; bet, $56.75; H, 168.75; T, $84.75.
TT, $56.25; bet, $28.125; H, $84.75; T, $42.1825;
Average: $142.382813.
Well, that's just what we expect: 100 +1.125 * 1.125 * 1.125 = $142.382813. No rebalancing bonus, just the intrinsic 12.5% per flip.
Here's where I think the mental problem comes: "For a player who gambles his entire bankroll each round, it appears to be a wash. No matter the order of returns, if there are an equal number of heads and tails, the player ends up having exactly as much as he did at the start." That makes it sound as if it is a wash for the player who gambles his entire bankroll each round if the coin flips are random, "since" by the law of averages there will be equal numbers of heads and tails.
That's completely wrong. The game in which the player bets his whole stake is not a wash; it is hugely favorable to the player. His expectation is an equal chance of having either $200 or $50, average $125. His mathematical expectation is to increase his holdings by 25% per flip. That would mean that the player who stakes all should do better than the player who stakes half, and this is true. Let's run through it again, including all combinations, not just those with equal numbers of heads and tails.
One flip:
$100 > H, $200, T, $50
Average, $125.
Two flips:
H, $200; H, $400, T, $100.
T, $50. H, $100, T, $25.
Average: $156.50.
The equal headsandtails runs left him back where he started, but because he won more on HH than he lost on TT, he made money overall. Indeed, he made $100 * 1.25 * 1.25 = $156.50.
For three flips, it works out to $195.625 but I'm not going to show that one.
I'm going to go out to four flips to get some of the equal head/tail runs.
HHHH $1,600.00
HHHT $400.00
HHTH $400.00
HTHH $400.00
THHH $400.00
HHTT $100.00
HTHT $100.00
HTTH $100.00
THHT $100.00
THTH $100.00
TTHH $100.00
HTTT $25.00
THTT $25.00
TTHT $25.00
TTTH $25.00
TTTT $6.25
Average: $244.14
Again, just what's expected: 100 * 1.25^4 = $244.14.
The boldfaced entries are those with equal numbers of heads and tails, and those are a wash for the player, but the game is hugely in his favor overall because the runs with more heads far more than outweigh the runs with more tails.
The hidden "reversion to mean" assumption comes from showing examples in which there were equal numbers of heads and tails. If you're guaranteed equal numbers of heads and tails, then betting half and rebalancing beats betting all each time. If, however, heads and tails are in fact random, then when you bet it all you earn 25% per flip on the average, and when you bet half you earn 12.5% per flip on the average.
The false logic is this:
a) If there are equal numbers of heads and tails, it's a wash for the player who bets it all, but the player who bets half comes out ahead and thus does better than the player who bets it all.
b) On the average, there will be equal numbers of heads and tails.
c) Therefore on the average the player who bets half will do better than the player who bets it all.
Last edited by nisiprius on Fri May 24, 2013 8:23 pm, edited 1 time in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
 Peter Foley
 Posts: 3178
 Joined: Fri Nov 23, 2007 10:34 am
 Location: Lake Wobegon
Re: "There is a rebalancing bonus ... false"
I believe there is a difference between theory and practice. If rebalancing helped someone stay the course, in essence keep from selling when the market dropped because they could not handle the risk, then it will have worked. There is no way I know of to model the influence of emotion with regard to rebalancing and overall investment returns.
Re: "There is a rebalancing bonus ... false"
The following statement summarizes everything that's wrong with this article:
Anybody who says that stocks were more risky at the end of 2008 than at, say, the beginning of 2008 should not be listened to, IMO.
No one reduced risk by exchanging debt for more stocks at 2008's year end  a time when stocks were the most risky they have been in a living memory.
Anybody who says that stocks were more risky at the end of 2008 than at, say, the beginning of 2008 should not be listened to, IMO.
Most of my posts assume no behavioral errors.
Re: "There is a rebalancing bonus ... false"
nisiprius wrote:rmelvey: I've figured out to my satisfaction, but probably not yours, why your "Shannon's Demon" example is misleading.
Nisiprius 
You are fundamentally missing the Shannon/Kelly point around geometric vs. arithmetic returns. Yes, betting the whole amount has the highest arithmetic return  but look at your distribution of returns (just after four flips!)  only 5 of 16 "portfolios" have any gain whatsoever. The "rebalancing bonus" is that if you bet 50%, you will have a much lower standard deviation, better sharpe ratio, and much better results across a broad range of outcomes (vs. your case that depends on one unlikely outcomes for the vast majority of the return). This is easy to model in excel  I've built a spreadsheet that does 10,000 trials of a 200 flip game (just completely random "coinflips", no mean reversion). The results are below:
You can decide for yourself if there's a rebalancing bonus and which portfolio you would prefer. If anyone would like the spreadsheet, just pm me I will send.
Re: "There is a rebalancing bonus ... false"
bmdaniel wrote:You can decide for yourself if there's a rebalancing bonus and which portfolio you would prefer. If anyone would like the spreadsheet, just pm me I will send.
Welcome! Alternatively, you can upload your spreadsheet to Google Drive and share the link. If Google corrupts the translation to its spreadsheet format, just upload the file without conversion.
 Rick Ferri
 Posts: 8154
 Joined: Mon Feb 26, 2007 11:40 am
 Location: Deep in the Heart of Texas
 Contact:
Re: "There is a rebalancing bonus ... false"
People often ask me if there is a best method for rebalancing. My answer is always the same. The best method is the one you’re able to implement religiously without emotion. After that, the method that results in the best riskadjusted return going forward is more luck than science.
Choices in Portfolio Rebalancing
Rick Ferri
Choices in Portfolio Rebalancing
Rick Ferri
Choose a few lowcost index funds in different asset classes, rebalance occasionally and forgetaboutit!
Re: "There is a rebalancing bonus ... false"
bmdaniel wrote:nisiprius wrote:rmelvey: I've figured out to my satisfaction, but probably not yours, why your "Shannon's Demon" example is misleading.
Nisiprius 
You are fundamentally missing the Shannon/Kelly point around geometric vs. arithmetic returns. Yes, betting the whole amount has the highest arithmetic return  but look at your distribution of returns (just after four flips!)  only 5 of 16 "portfolios" have any gain whatsoever. The "rebalancing bonus" is that if you bet 50%, you will have a much lower standard deviation, better sharpe ratio, and much better results across a broad range of outcomes (vs. your case that depends on one unlikely outcomes for the vast majority of the return). This is easy to model in excel  I've built a spreadsheet that does 10,000 trials of a 200 flip game (just completely random "coinflips", no mean reversion). The results are below:
You can decide for yourself if there's a rebalancing bonus and which portfolio you would prefer. If anyone would like the spreadsheet, just pm me I will send.
But is the assumption of memoryless financial markets reasonable?
Most of my posts assume no behavioral errors.
Re: "There is a rebalancing bonus ... false"
baw703916 wrote:
But is the assumption of memoryless financial markets reasonable?
I was primarily responding to the comment that mean reversion is required for the Shannon point to hold (I.e. the value of the Kelly criterion where applicable). Much more complicated question how that translates into the impact of rebalancing among extremely broad index fund choices, for example. Would be very hesitant to say it has no market relevance though, see E.O Thorp for example.
Will cleanup the spreadsheet and try to post later.
Re: "There is a rebalancing bonus ... false"
Spreadsheet available at the below link:
https://docs.google.com/file/d/0B38C_Am ... sp=sharing
Would recommend turning calculation to manual before using, there are 2 million coin flips happening every time it calcs.
https://docs.google.com/file/d/0B38C_Am ... sp=sharing
Would recommend turning calculation to manual before using, there are 2 million coin flips happening every time it calcs.
Re: "There is a rebalancing bonus ... false"
The file is 84.6 MB. Even with recalculation disabled, it's taking far too long to load and I ended up aborting. Can you reduce the size, such as putting the number of coin flips in a cell, setting it to 1, then let me change it to 2e6? I'm using LibreOffice Calc, but that shouldn't matter.
 nisiprius
 Advisory Board
 Posts: 30778
 Joined: Thu Jul 26, 2007 9:33 am
 Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.O. Henry
Re: "There is a rebalancing bonus ... false"
Look, someone had better state clearly and unambiguously what the proposition is, because what tends to happen these discussions is that different advocates of rebalancing put forward different claims for what it is supposed to be doing, never quite clearly stated.bmdaniel wrote:nisiprius wrote:rmelvey: I've figured out to my satisfaction, but probably not yours, why your "Shannon's Demon" example is misleading.
Nisiprius 
You are fundamentally missing the Shannon/Kelly point around geometric vs. arithmetic returns....
I am perfectly clear on the concept that of there being a riskreward proposition in deciding whether to rebalance or not. Obviously the investor who does not rebalance will experience both more risk and higher expected returns than one who does. As far as I'm concerned, the reason for rebalancing is simply that you don't want to exceed your risk tolerance and don't want your asset allocation to get wildly out of whack with what you planned.
I was specifically addressing rmelvey's presentation here, and I am waiting to see if he responds, but his presentation has nothing to do with Sharpe ratios or anything like that. He presents the proposition as "appearing to be a wash" for the gambler who bets everything each time. I read him, perhaps incorrectly, as implying that this really is true for the gambler who bets everything each time; for that strategy, he seems to imply that the game really is a wash. Is says the game is "an incredible profit opportunity," only for the gambler "sophisticated" enough to bet half each time instead of allto use a "rebalanced portfolio of 50% coin flip and 50% cash."
Well, this is wrong. The game is an incredible profit opportunity, period. The 50/50 investor increases the returns in the maximumlikelihood case (equal numbers of heads and tails) and generally reshapes the distribution of outcomes to something that would certainly be closer to my liking. However, it's equally clear that betting it all each time produces so much higher results in some of the less likely cases that it more than overbalances the maximumlikelihood case. To put it plainly, for the gambler who bets everything game generates an expected 25% yield per flip, no matter what. For the gambler who bets half, the game generates an expected 12.5% yield per flip, no matter what.
It's also utterly irrelevant to anything about realworld investing because the example is so bizarrely different from anything that could exist in the real world that it hardly repays thinking about the relative behavior of different strategies. When someone actually offers me an opportunity with a return expectation of 25% per coin flip, I will examineno I won't, I will run, because it is clearly a proposition too good to be true.
The thing about arithmetic and geometric means is really simple: for any unit time period, the right thing to do is to take the arithmetic means of the returns of the assets in the portfolio; for periods spanning more than one unit, you take the geometric mean of those arithmetic means.
Now, all this stuff, like the Monty Hall problem and so many things in probability, is tricky enough to think about that it is genuinely hard to be sure you've got it right, particularly if it's offered up in some new guise, hence the endless debate.
There are a fair number of people, though, who really believe that rebalancing magically ratchets up returns because it is an automatic mechanical way of buying low and selling high, and that the effect depends simply on the fact that it is rebalancing, and doesn't depend on mean reversion. As opposed to saying that if you rebalance at periodic intervals, you may actually catch a small benefit if the rebalancing interval happens to resonate with the mean reversion period.
No mean reversion, no rebalancing bonus. You always have the opportunity to shape the distribution of outcomes and remold it nearer to the heart's desire, but which shape is best depends on one's personal heart's desirethat is to say, one's appetite for risk and which direction one prefers to have the outcomes skewed.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: "There is a rebalancing bonus ... false"
I agree, as I mentioned above that it's not particularly applicable to rebalancing massive market wide index funds, because you are getting so much rebalancing underneath as well that the majority of the "risk of ruin" is already removed. If you were putting on a handful of particular trades/investments, it's more likely to be meaningful. However, I'm not convinced that it wont be a bit more broadly applicable than that (see my comment in the commodity thread, for example).
That said, I think it's a bit disingenuous to say it's just a "reshaping" to go from a distribution where you have a ~50% chance of making nothing to a distribution with an almost certainty of making money for a minuscule reduction in average return. It's basically the St. Petersburg paradox  you are not going to pay a million dollars to play, even though your expected return would be essentially infinite. If you want a really welldone concise description of the issue, I highly recommend Fortune's Formula by William Poundstone.
That said, I think it's a bit disingenuous to say it's just a "reshaping" to go from a distribution where you have a ~50% chance of making nothing to a distribution with an almost certainty of making money for a minuscule reduction in average return. It's basically the St. Petersburg paradox  you are not going to pay a million dollars to play, even though your expected return would be essentially infinite. If you want a really welldone concise description of the issue, I highly recommend Fortune's Formula by William Poundstone.
Re: "There is a rebalancing bonus ... false"
LadyGeek wrote:The file is 84.6 MB. Even with recalculation disabled, it's taking far too long to load and I ended up aborting. Can you reduce the size, such as putting the number of coin flips in a cell, setting it to 1, then let me change it to 2e6? I'm using LibreOffice Calc, but that shouldn't matter.
Have replaced with a version that only runs 1/10th the trials (and so is a 10th the size). The average mean will be a bit jumpier (b/c as discussed it's very levered to extremely unlikely cases); recommend refreshing a few times so you can see where the various values settle out. Medians and percent of losing cases should be very steady.
Re: "There is a rebalancing bonus ... false"
^^^ 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.
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.
 nisiprius
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 Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.O. Henry
Re: "There is a rebalancing bonus ... false"
I'd still like you to state, with precision, what it is that you think rebalancing accomplishes, in the absence of mean reversion. I do not mean that it does nothing. I want a correct description of what it does do.bmdaniel wrote:I agree, as I mentioned above that it's not particularly applicable to rebalancing massive market wide index funds, because you are getting so much rebalancing underneath as well that the majority of the "risk of ruin" is already removed. If you were putting on a handful of particular trades/investments, it's more likely to be meaningful. However, I'm not convinced that it wont be a bit more broadly applicable than that (see my comment in the commodity thread, for example).
That said, I think it's a bit disingenuous to say it's just a "reshaping" to go from a distribution where you have a ~50% chance of making nothing to a distribution with an almost certainty of making money for a minuscule reduction in average return. It's basically the St. Petersburg paradox  you are not going to pay a million dollars to play, even though your expected return would be essentially infinite. If you want a really welldone concise description of the issue, I highly recommend Fortune's Formula by William Poundstone.
For example, an incorrect description would be "increase your returns by automatically making you buy low and sell high."
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
Re: "There is a rebalancing bonus ... false"
An interesting reversion to mean concept is that, RTM will occur almost certainly (and maybe "certainly", but I dunno if there are discontinuous, imaginary, or extreme regular cases where it could be said not to occur), but that we just do not know what the mean will be.....
http://www.norstad.org/finance/rtmand ... g.html#rtm
so yeah, ex post, there is "mean reversion", but going forward, one doesnt know the mean.
http://www.norstad.org/finance/rtmand ... g.html#rtm
We see stock market charts. Our eye draws the line from the starting point to the ending point. We notice that the chart goes up and down, but eventually it always comes back to that nice straight line in the middle of all the jagged ups and downs. Our common sense mistakenly calls this "mean reversion," and we think we are seeing something significant, when what we are really seeing is just a useless triviality (what we are seeing is an immediate consequence of the definition of "average"  if you take an average of things, some of the things are above the average, and some are below, and that information is not of much significance or use).
so yeah, ex post, there is "mean reversion", but going forward, one doesnt know the mean.
Re: "There is a rebalancing bonus ... false"
I don't know if it does anything in the context of rebalancing stock/bond allocations; evidence pretty unconvincing either way. As I said before, this rebalancing is likely dwarfed by the underlying diversification.
In certain situations however, it can significantly increase risk adjusted return while minimizing "risk of ruin" (the most important consideration in a longterm geometric return scenario). I do believe this applies to some market investing, such as concentrated hedge fund type strategies. Again, I was primarily responding to your claim that you had figured out the logical flaw in the Shannon example, which I disagreed with  rebalancing in this case certainly doesn't rely on mean reversion.
In certain situations however, it can significantly increase risk adjusted return while minimizing "risk of ruin" (the most important consideration in a longterm geometric return scenario). I do believe this applies to some market investing, such as concentrated hedge fund type strategies. Again, I was primarily responding to your claim that you had figured out the logical flaw in the Shannon example, which I disagreed with  rebalancing in this case certainly doesn't rely on mean reversion.
Re: "There is a rebalancing bonus ... false"
boggler wrote:I have seen arguments that rebalancing does have a bonus if the market exhibits momentum, but I'm not sure how long a momentumbased strategy will last in the age of highfrequency trading. For those that do believe there is a rebalancing bonus, how do you justify it?
Actually, if the market exhibits momentum rebalancing can be a loser. This is an argument against rebalancing too often if there is short term momentum.
If the market is a randomwalk, then it has no memory (it's next move is not a function of its history) so there is not way that rebalancing can work, since it's based on the memory that it moved away from your AA. "Buy low and sell high" is based on memory that the market is high or low relative to the past, and a random walk does not respond to that.
I works if (relative to each other) return rates of your assets are going to be lower in the future because is was higher in the past and visa versa.
Re: "There is a rebalancing bonus ... false"
I don't understand all the chatter about mean reversion. Rebalancing bonus / diversification benefit is based on low or negative correlation of portfolio components. Anyway, the real choices are 1) to reduce risk vs. letting things ride and 2) to construct a portfolio with components that show (at least historical) low/negative correlation to smooth volatility / returns or not (?).

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 Joined: Sat May 25, 2013 8:19 pm
Re: "There is a rebalancing bonus ... false"
Nisiprius wrote, with reference to rmelvey's coin flip game: "The game is an incredible profit opportunity, period." This is certainly true if many such games are played in parallel. For example, suppose you buy 1000 $1 tickets, each good for one coin flip. Each winning ticket generates a dollar of profit; each losing ticket a 50 cent loss. Hence, a winning ticket makes up for two losing tickets. You'll make an overall profit as long as more than 1/3 of your tickets are winners. With 1000 tickets the chance of at least 1/3 winners is close to 1 (less than 1 chance in 100 million of fewer than 1/3 winners), so an overall profit is all but guaranteed. Roughly 99% of the time the number of winning tickets will be in the range 460540 (you can use any of several online calculators for the cumulative binomial distribution to derive these numbers). Hence, roughly 99% of the time your winnings will be in the range $190$310, since 460 winning tickets means $460 of gross profits and $270 = 0.5*($1000$460) of gross losses, for a net profit of $190, and similarly for the case of 540 winning tickets. In the parallel case expected wealth depends on the arithmetic mean of the outcomes. Here, the expected outcome is 25% profit, equivalent to winning exactly 500 games, generating $500 of gross profit and $250 of gross loss, for a net profit of $250.
Things are quite different if many such games are played in sequence, with reinvestment, in full or in part, of one's accumulated stake. In this case the returns accumulate multiplicatively, not additively. With full reinvestment, a winning ticket only makes up for a single losing ticket, not two losers as in the parallel case (this is an example of the oftrepeated observation that you need 100% growth to make up for a 50% loss). We now need more than 50% winners to generate an overall profit, and this will only happen half of the time. In the long run we expect as many losing tickets as winners, so our longterm expectation is a flat equity curve. Suppose we play 1000 sequential games with full reinvestment. As above, roughly 99% of the time we will have between 460 and 540 winners. We've excluded the 1% most extreme outcomes but are still left with a gargantuan range of possible values for our terminal wealth as a multiple of our initial investment, from vanishingly small (less than one septillionth) to astronomically large (more than a septillion). Yes, the arithmetic mean of these outcomes is also astronomically large, since half the outcomes are quite small (in the range [0, 1]) and half are large to enormous (in the range [1, 10^24]). However, for any x, we are just as likely see our stake reduced to 1/x of its starting value as we are to see it multiplied to x times its starting value, so the arithmetic mean is not a useful guide to our expected terminal wealth. While a single sequence of games has only a 50% chance of turning a profit, playing many sequences in parallel will generate fabulous wealth with only a vanishingly small chance of loss (just apply the logic from the first paragraph to the new distribution of outcomes generated in this paragraph for sequences of coin flips).
Now consider 1000 sequential games with 50% reinvestment. A winning ticket multiplies our ongoing stake by 1.5, a losing ticket by 0.75. Therefore, a winning ticket more than makes up for a single losing ticket (a 25% loss only requires 33.3% growth for a full recovery and a winning ticket supplies 50% growth). In fact, 5 winning tickets cancel out 7 losing tickets with some leftover spare profit. 99.5% of the time we expect to get at least 460 winning tickets. Therefore, over 1000 plays, our wealth will grow by at least a factor of 34 trillion (= 1.5^460 * 0.75^540) about 99.5% of the time. If allowed to invest in only a single sequence of games, I would certainly opt for 50% reinvestment over 100%.
In the sequential case one's expected wealth grows as the geometric mean of the possible outcomes. This is sqrt(2*0.5) = 1 in the full reinvestment case, meaning no expected growth. In the 50% reinvestment case the geometric mean is
sqrt(1.5*0.75) = sqrt(1.125) = 1.06, i.e., we can expect our wealth to grow, in the long run, 6% per flip. The arithmetic mean is higher with 100% reinvestment but volatility drags the expected growth down to 0. Reinvestment at 50% reduces the arithmetic mean but reduces the volatility drag even more, producing positive expected growth.
Are there any applications of these ideas to real investments? An investment of $1000 in EEM, the emerging markets ETF, made at the close of 12/31/2005 would be worth $1222 today (5/25/2012), assuming all dividends were reinvested. A portfolio that was constantly rebalanced to keep 75% in EEM and 25% in cash under the mattress would be worth $1272 today: a 4.1% rebalancing bonus (despite the 0% return on mattress cash) along with 25% less volatility.
Things are quite different if many such games are played in sequence, with reinvestment, in full or in part, of one's accumulated stake. In this case the returns accumulate multiplicatively, not additively. With full reinvestment, a winning ticket only makes up for a single losing ticket, not two losers as in the parallel case (this is an example of the oftrepeated observation that you need 100% growth to make up for a 50% loss). We now need more than 50% winners to generate an overall profit, and this will only happen half of the time. In the long run we expect as many losing tickets as winners, so our longterm expectation is a flat equity curve. Suppose we play 1000 sequential games with full reinvestment. As above, roughly 99% of the time we will have between 460 and 540 winners. We've excluded the 1% most extreme outcomes but are still left with a gargantuan range of possible values for our terminal wealth as a multiple of our initial investment, from vanishingly small (less than one septillionth) to astronomically large (more than a septillion). Yes, the arithmetic mean of these outcomes is also astronomically large, since half the outcomes are quite small (in the range [0, 1]) and half are large to enormous (in the range [1, 10^24]). However, for any x, we are just as likely see our stake reduced to 1/x of its starting value as we are to see it multiplied to x times its starting value, so the arithmetic mean is not a useful guide to our expected terminal wealth. While a single sequence of games has only a 50% chance of turning a profit, playing many sequences in parallel will generate fabulous wealth with only a vanishingly small chance of loss (just apply the logic from the first paragraph to the new distribution of outcomes generated in this paragraph for sequences of coin flips).
Now consider 1000 sequential games with 50% reinvestment. A winning ticket multiplies our ongoing stake by 1.5, a losing ticket by 0.75. Therefore, a winning ticket more than makes up for a single losing ticket (a 25% loss only requires 33.3% growth for a full recovery and a winning ticket supplies 50% growth). In fact, 5 winning tickets cancel out 7 losing tickets with some leftover spare profit. 99.5% of the time we expect to get at least 460 winning tickets. Therefore, over 1000 plays, our wealth will grow by at least a factor of 34 trillion (= 1.5^460 * 0.75^540) about 99.5% of the time. If allowed to invest in only a single sequence of games, I would certainly opt for 50% reinvestment over 100%.
In the sequential case one's expected wealth grows as the geometric mean of the possible outcomes. This is sqrt(2*0.5) = 1 in the full reinvestment case, meaning no expected growth. In the 50% reinvestment case the geometric mean is
sqrt(1.5*0.75) = sqrt(1.125) = 1.06, i.e., we can expect our wealth to grow, in the long run, 6% per flip. The arithmetic mean is higher with 100% reinvestment but volatility drags the expected growth down to 0. Reinvestment at 50% reduces the arithmetic mean but reduces the volatility drag even more, producing positive expected growth.
Are there any applications of these ideas to real investments? An investment of $1000 in EEM, the emerging markets ETF, made at the close of 12/31/2005 would be worth $1222 today (5/25/2012), assuming all dividends were reinvested. A portfolio that was constantly rebalanced to keep 75% in EEM and 25% in cash under the mattress would be worth $1272 today: a 4.1% rebalancing bonus (despite the 0% return on mattress cash) along with 25% less volatility.
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