POLL: How much re-balancing bonus do you expect?

[grayfox]

The basic Boglehead is to pick a target asset allocation and then stick with it. But over time, portfolio asset weights will drift from the target. At some point, re-balance back to target. This could be done periodically, like every year, month or day, or by threshold bands, like +/-5% or 10%.

For example, if the target portfolio had two funds, Total Stock Market (VTI) and Total Bond Market (BND), and the target was equal weighting, 50/50. Over time, this may drift to 60/40. If you had 10% rebalancing bands, you might rebalance back to 50/50, or maybe partway to 50/50. There's a million different re-balancing schema.

Some refer to a "re-balancing bonus", or re-balancing return, which would be the return from selling the winners and buying the losers.

How much extra return do you expect to get from re-balancing? Note that the poll answers allow for both positive (rebalancing bonus) and negative (rebalancing penalty) responses. If you like, feel free to describe your re-balancing method and explain why you answered the way you did.

[Noobvestor]

I went with 0 to negative 1, but it could be much worse (more negative) than that. If I really never rebalanced back from equities, I imagine my equity-heavier-and-heavier portfolio will more likely than not do much better than my balanced stock/bond one.

As for me: I rebalance with new money in fits and starts as it comes in, and may or may not if I need to sell things to do it, since a lot of my port is taxable. In theory, I have bands. In practice, I have tax consequences. I haven't hit a problem yet that couldn't be solved with new money added, but may at some point, so we'll see.

[Random Musings]

Equities, in the long-run, have outperformed bonds, and the expectations are that they will continue to do so. Hence, on average, you are rebalancing more often from equities to bonds which should diminish returns.

However, if one looks at it from a set allocation point (say 50-50 target), it's not the "rebalancing bonus" that is the driver, IMHO. It's more about sticking with your portfolio need of risk in your written investment plan.

RM

[jidina80]

I voted 'zero to negative one' because most of my rebalancing over the past 25 years has been from stocks to bonds. Since stocks tend to perform better than bonds over the long run, I probably would have had a higher return, but riskier portfolio, if I left the money in stocks. I rebalance to manage risk.

However, when rebalancing between U.S. and foreign stocks, I would expect a 'zero to positive one' percent gain due to the 'buy low, sell high' tendency.

[livesoft]

I expect a bonus of between 1% and 2%. I do not rebalance based on the calendar nor do I use 5% to 10%. Instead, I use the RBD method which seems to work extremely well.

[grayfox]

I voted Approximately 0% as the expected re-balancing return.

Now, I have no doubt that if there are two portfolios, one rebalanced and one not, that they will end up with different results. I don't believe the rebalancing return ex-post will be zero. But I think the rebalancing return, which I would characterize as a return from a trading strategy, can be positive or negative, and it depends on luck and timing.

I found a paper by Vanguard, Portfolio Rebalancing in Theory and Practice that looked into re-balancing. They conclude that rebalancing subtracts from returns in trending markets and adds to returns in mean-reverting markets. If you are in a trending market, the rebalancing strategy that rebalances less often will have higher rebalancing return. For example,10% bands instead of 5%. But if you wait too long before rebalancing and the trend reverses, then you missed out. It all seems to be in the timing.

[grayfox]

I expect a bonus of between 1% and 2%. I do not rebalance based on the calendar nor do I use 5% to 10%. Instead, I use the RBD method which seems to work extremely well.

It doesn't look like there are any votes for 1 to 2%

[livesoft]

^ I didn't say I voted, did I?

[grayfox]

One of the reasons I created this poll was that I saw this old thread Opportunistic Rebalancing: A New Paradigm from 2008 had been revived

Now if you study Modern Finance in university, one of the results is that return of a portfolio is equal to the weighted sum of the returns of the components.

Rp = w1*R1 + w1*R2 + w3*R3 + ... wn*Rn

As far as I have seen up to this point, balancing return is not part of the modern finance theory. They don't add in rebalancing bonus to Rp. When you solve for the efficient frontier (EF), the highest returning portfolio is 100% of the highest returning asset. The lowest returning portfolio is 100% of the lowest returning asset.

There can be portfolios on the EF that have lower volatility than the least volatile asset. This seemingly magical result is from uncorrelated assets. But there are no portfolios on the efficient frontier that have higher return than the highest return asset.

That's why I would call the rebalancing return profit & loss from trading.

[Noobvestor]

One of the reasons I created this poll was that I saw this old thread Opportunistic Rebalancing: A New Paradigm from 2008 had been revived

Now if you study Modern Finance in university, one of the results is that return of a portfolio is equal to the weighted sum of the returns of the components.

Rp = w1*R1 + w1*R2 + w3*R3 + ... wn*Rn

As far as I have seen up to this point, balancing return is not part of the modern finance theory. They don't add in rebalancing bonus to Rp. When you solve for the efficient frontier (EF), the highest returning portfolio is 100% of the highest returning asset. The lowest returning portfolio is 100% of the lowest returning asset.

There can be portfolios on the EF that have lower volatility than the least volatile asset. This seemingly magical result is from uncorrelated assets. But there are no portfolios on the efficient frontier that have higher return than the highest return asset.

That's why I would call the rebalancing return profit & loss from trading.

That's a really interesting result. I almost wonder if the next poll should be something like: do you expect the bond component of your portfolio to increase returns? I'd never presumed it might in my case, but figured it could actually do that in the case of someone with 80-90% stocks, say, and long-term bonds - i.e. I thought perhaps, without ever bothering to run numbers, that some portfolios might indeed beat their top-asset return if they had something volatile enough to facilitate rebalancing at extreme points.

[maddyken]

I voted approximately zero.

I have doubts about rebalancing back to a DIY AA. Is there any reason to believe an initially arbitrary AA is efficient or will be efficient over the long run?

[Random Walker]

GrayFox,
I have not taken any university level finance classes. But based on my reading of Boglehead books, I think I disagree with what you posited. The return of a portfolio cannot simply be the weighted average return of the components. The compounded return of the portfolio is always less than the weighted average annual return of the components due to portfolio volatility. In fact, the more you can dampen portfolio volatility through investing in multiple asset classes, the closer the compounded return will be to the average annual return of the components. The positive effect of dampened portfolio volatility on annualized (compounded) returns is what I would view as the rebalancing bonus.
The potential benefit of adding an asset class to a portfolio depends on more than its return. It depends on expected return, correlation to other portfolio components, volatility. Curious to hear your thoughts.

Dave

[mickeyd]

At this point I will not be making any rebalancing decisions now. My IPS allows me to make rebalancing decisions up to 1 quarter past the end of the year (Mar 31) if rebalancing is not indicated at the end of the year. I will be in postponement mode from Jan-Mar31, just like last year.

[maddyken]

I agree with GrayFox.

[maddyken]

The ideal is to build a portfolio which will deliver, say, 8% per year with zero volatility. Then your planning is straightforward.

In the real world the portfolio SD is going to cause the portfolio return to fall below 8%, so we try to minimize deviations from 8% in order to keep our plan on track.

But there's no way a portfolio made up of a 6% class and an 8% class is going to have higher portfolio return than a portfolio made up of only the 8% class. I'm looking backward here.

That said, it is possible the 6/8 combo has better risk-adjusted return but it will never equal the 8% performance of the single class portfolio.

[maddyken]

Doing a full backward-looking mean-variance optimization the only inputs are the returns and the weights for each investment. The interaction among the classes/investments is derived from the returns.

In a roundabout way you can see why backward-looking analyses have significant limitations.

That's not to say crystal balls don't have limitations, but...

[maddyken]

The portfolio return is the sum of the weighted returns of the individual classes.

The returns of the individual classes are determined at the end of the period, after each class SD has been accounted for (rolled up into the class return)...and consequently rolled up into the portfolio return. IOW, for return purposes, the SD has already been accounted for and is thus irrelevant...that's why all the interactive stuff doesn't matter when calculating portfolio return.

I think everything you stated after your denial of how portfolio return is determined is basically true, if you step back and look at everything after the fact you'll see what I mean.

Gratis.

[maddyken]

Are you still with me?

[maddyken]

I guess I have to realize I can dump on presentations and interpretations in uncomfortable ways, always to the benefit of the readers.

Good luck to all, and if that means me learning a lesson or two so be it.

[ourbrooks]

The compounded return of the portfolio is always less than the weighted average annual return of the components due to portfolio volatility.
Dave

Not true. Suppose I have two investments that have an expected return of 8% but one has twice the volatility of the other. I open two accounts, one with each investment, and wait long enough, say, 30 or 40 years. Both accounts will be up 8%; that's what expected return means. In the meantime, one account will have moved up and down twice as much as the other, but the end result will be the same.

Now let's suppose that the investments both have really high volatility but they happen to have a 1.0 negative correlation with each other so that when one zigs the other zags. At the end of the long time period, both investments will be up 8%, no more no less, but if I plot the total of the two investments for each year, it will be a nice smooth straight line instead of moving up and down on either side of the 8% line.

Volatility has no effect on the expected value of investments over a long term; Grayfox is absolutely correct. On the other hand, reducing volatility means that any year's returns will be closer to the long term expected value but the expected value itself won't change.

[Dale_G]

I have run the numbers on the effect of rebalancing my accounts from 2002 through mid 2012. It can be a bit tedious accounting for dividends, but I am quite confident of the numbers. The rebalancing benefit averaged about 0.73% of the portfolio value per year based on round trip transactions. My asset allocation was 50/50 and I maintained fairly tight rebalancing bands.

It was not much fun rebalancing into the decline of 2008/2009, but the "bonus" eventually realized made up for some of the discomfort.

Volatility can be your friend, but only if you rebalance.

Dale

[livesoft]

I wonder what the bonus would be for someone who rebalanced on the best day for rebalancing of the year each year. Would that not set an upper bound for a rebalancing bonus for the once-a-year rebalancer? Would that not be a number to strive for? I suppose it might depend on stock:bond ratio as well.

[richard]

I have run the numbers on the effect of rebalancing my accounts from 2002 through mid 2012. It can be a bit tedious accounting for dividends, but I am quite confident of the numbers. The rebalancing benefit averaged about 0.73% of the portfolio value per year based on round trip transactions. My asset allocation was 50/50 and I maintained fairly tight rebalancing bands.

Rebalancing does better when you have two volatile assets with similar returns, as we had with stocks and bonds over the past ten years. If stocks do substantially better than bonds, rebalancing is likely to lower returns.

[grayfox]

GrayFox,
I have not taken any university level finance classes. But based on my reading of Boglehead books, I think I disagree with what you posited. The return of a portfolio cannot simply be the weighted average return of the components. The compounded return of the portfolio is always less than the weighted average annual return of the components due to portfolio volatility. In fact, the more you can dampen portfolio volatility through investing in multiple asset classes, the closer the compounded return will be to the average annual return of the components. The positive effect of dampened portfolio volatility on annualized (compounded) returns is what I would view as the rebalancing bonus.
The potential benefit of adding an asset class to a portfolio depends on more than its return. It depends on expected return, correlation to other portfolio components, volatility. Curious to hear your thoughts.

Dave

What you are talking about is that the arithmetic mean is less than the geometric mean when there is volatility. That is true even for one asset. For example, if you have twelve monthly returns, take the arithmetic mean, and get 5%. The geometric mean will be less than 5%, unless every month was exactly 5%, i.e. no volatility. Any variation will give lower geometric return.

And, yes, the theory is to minimize volatility at some return. Calculate the minimum volatility across the range of returns, and that is the efficient frontier. The surprising thing is that you can get portfolio with lower volatility than the least volatile asset. So 10/90 stocks/bonds might be lower volatility than 0/100, due to lack of correlation.

[grayfox]

I have run the numbers on the effect of rebalancing my accounts from 2002 through mid 2012. It can be a bit tedious accounting for dividends, but I am quite confident of the numbers. The rebalancing benefit averaged about 0.73% of the portfolio value per year based on round trip transactions. My asset allocation was 50/50 and I maintained fairly tight rebalancing bands.

It was not much fun rebalancing into the decline of 2008/2009, but the "bonus" eventually realized made up for some of the discomfort.

Volatility can be your friend, but only if you rebalance.

Dale

I calculated the return 1) without re-balancing and 2) with daily re-balancing for a Harry Browne portfolio for 2011. The portfolio with daily re-balancing had 0.89% higher return. But I only looked at one year, nor did I look at other re-balancing rules.

The Vanguard paper found that in trending markets you get a negative re-balancing return and in mean reverting markets you get a positive rebalancing return. If you went from 2007 to 2010, there was some mighty mean reversion.

As I said earlier, I have no doubt that, ex-post, you can measure a positive rebalancing return if you time it well. But that has nothing to do with portfolios. You can get it with one security. Suppose IBM returned 10% in 2012. Well if it fluctuated in price, I might have got 12% from that stock just by timing my transactions. I could have sold when it rose, and bought back more shares when it fell. Or even without ever selling, if I'm added new contributions, just buying on dips.

I just don't know if you can say in advance if you will see a bonus or a penalty from re-balacning.

[mlewis]

Equities, in the long-run, have outperformed bonds, and the expectations are that they will continue to do so. Hence, on average, you are rebalancing more often from equities to bonds which should diminish returns.

However, if one looks at it from a set allocation point (say 50-50 target), it's not the "rebalancing bonus" that is the driver, IMHO. It's more about sticking with your portfolio need of risk in your written investment plan.

RM

Diminish returns as compared to what? It is true that the un-rebalanced portfolio could do better than the rebalanced one, but only at the expense of increased risk. A rebalanced portfolio can still beat the arithmetic mean return of the un-rebalanced portfolio.

Here is a quote from an old EF article.
"Only when long term return differences among asssets exceed 5 percent do nonrebalanced portfolios provide superior returns, and then only at the cost of increased risk"

I recommend it:
http://www.efficientfrontier.com/ef/197/rebal197.htm
and:
http://www.efficientfrontier.com/ef/996/rebal.htm
http://www.efficientfrontier.com/ef/797/rebal797.htm

malcolm

[Random Walker]

I'm not really buying what I'm reading here. I certainly could be wrong. But it seems to me that the return of a REBALANCED portfolio cannot simply be the weighted average of the individual components. One needs to look at dollar weighted returns, not simply time weighted. That is the whole potential benefit of rebalancing: buying relatively low and selling relatively high at the asset class level.
If I am wrong, please continue to convince me. I want to understand this stuff. So far though I'm sticking with compounded return less than average annual. And the goal of multi-asset class investing with rebalancing is to get the annualized return closer to the average annual return. But need to look at dollar weighted, not just time weighted. Thanks.

Dave

[umfundi]

Oh, my! Rebalancing into a trending market. Rebalancing into a reverting market.

When will we ever run out of lipstick for the pig of market timing?

Over the last decade, suppose you invested the same amount monthly into a 50/50 mix of stocks and bonds.

1. You invested and did not rebalance.
2. You rebalanced monthly to keep your allocation at 50/50.

Option 2, continuous rebalancing, is superior to the tune of about 0.3% per year.

In my view, you may never rebalance, and perhaps see your portfolio AA drift to higher returns (and higher risks). Or, you may continually rebalance and see a lower risk and probably better returns.

Anything in between is market timing.

Keith

[Aptenodytes]

Rebalancing, for me, is a risk management strategy, not an optimization strategy. If I don't rebalance my risk exposure gets off. So the more relevant question for me, not necessarily everyone, is how much lower is the probability of a catastrophic loss if I rebalance? I see that benefit as very large.

I realize risk and return are two blades of the same scissors, but when I rebalance I am thinking risk more than return. Seems at a minimum you'd want to look at the return bonus and the risk benefit, not just one.

In any event the counterfactual is a bit bizarre. Having chosen an AA why wouldn't you rebalance to maintain it?

[grayfox]

I'm not really buying what I'm reading here. I certainly could be wrong. But it seems to me that the return of a REBALANCED portfolio cannot simply be the weighted average of the individual components.

Dave

That's exactly what I said. Rebalanced portfolio, by buying and selling, will end up with a different return than simply holding a portfolio. It could be higher or lower. In other words, the rebalancing return can be positive or negative.

But it has nothing to do with modern portfolio theory. It is explained by old fashioned buying low and selling high.
If stocks go lower after you sell stocks to rebalance, you will have higher return than holding. <- rebalancing bonus
If stocks continues higher after you sell to rebalance, you will have lower return than holding. <- rebalancing penalty

Here is a link to Modern portfolio theory. You will not see anything about a rebalancing bonus on that page. Below is the formula for expected return from that page, the weighted sum of the components.

[grayfox]

Rebalancing, for me, is a risk management strategy, not an optimization strategy. If I don't rebalance my risk exposure gets off. So the more relevant question for me, not necessarily everyone, is how much lower is the probability of a catastrophic loss if I rebalance? I see that benefit as very large.

I realize risk and return are two blades of the same scissors, but when I rebalance I am thinking risk more than return. Seems at a minimum you'd want to look at the return bonus and the risk benefit, not just one.

In any event the counterfactual is a bit bizarre. Having chosen an AA why wouldn't you rebalance to maintain it?

That's the bigger picture.

If I have a target portfolio that is 30/70, then it make sense to maintain that by rebalancing. For whatever reasons, I chose 30/70, and it wouldn't make sense to have 20/80, 50/00, 0/100 or anything else. Ideally, that would mean rebalancing as often as practical. With mutual funds that might be every two months because of frequent trading rules.

But it seems that some are advising, "No, delay rebalancing for a couple of years. Then you will get the maximum re-balancing bonus." I can cite books that recommend that. I've also read advice to choose 50/50 asset allocation because it will maximize the re-balancing bonus.

So the poll is to see how many here believe that they can get a re-balancing bonus by timing their re-balancing moves. From the poll, it looks like over 40% of respondents believe that they will get a rebalancing bonus by well-timed re-balancing.

BTW, the poll is set up so that you can change your vote, if anyone is having second thoughts about the re-balancing bonus.

[umfundi]

So the poll is to see how many here believe that they can get a re-balancing bonus by timing their re-balancing moves.

If they're that smart, why don't they unbalance?

Keith

[Random Walker]

Gray fox,
Thanks for the comments. I'm sure I'm going to learn something from this. But I'm still disagreeing. The equation you post is the expected return ex ante of a portfolio comprised of individual components. It does not account for potential affects of internal rebalancing. I agree that if you start at time point 1 and finish at time point 2, then the portfolio return of a portfolio with no additions/subtractions and no rebalancing will be the weighted average of the components.
But I believe portfolio volatility will have a huge effect on compounded annualized returns and that volatility will be dampened by multi asset class investing. So I guess what I'm saying is the following. Multi asset class investing will dampen portfolio volatility and bring compounded return closer to average annual return. Rebalancing will maintain the risk profile and can have positive or negative effect on returns depending on the course of market, momentum, reversion to mean. Ok now decipher this and straighten me out. Thanks.

Dave

[grayfox]

Gray fox,
Thanks for the comments. I'm sure I'm going to learn something from this. But I'm still disagreeing. The equation you post is the expected return ex ante of a portfolio comprised of individual components. It does not account for potential affects of internal rebalancing. I agree that if you start at time point 1 and finish at time point 2, then the portfolio return of a portfolio with no additions/subtractions and no rebalancing will be the weighted average of the components.
But I believe portfolio volatility will have a huge effect on compounded annualized returns and that volatility will be dampened by multi asset class investing. So I guess what I'm saying is the following. Multi asset class investing will dampen portfolio volatility and bring compounded return closer to average annual return. Rebalancing will maintain the risk profile and can have positive or negative effect on returns depending on the course of market, momentum, reversion to mean. Ok now decipher this and straighten me out. Thanks.

Dave

I don't disagree with anything you just wrote. So I'm not sure what you disagreeing about. Are you disagreeing that we agree?

[livesoft]

So the poll is to see how many here believe that they can get a re-balancing bonus by timing their re-balancing moves.

If they're that smart, why don't they unbalance?

Keith

They do. On RBDs. They get a better return than Wellington, but have more bonds in their portfolios.

[ourbrooks]

Look at the Vanguard paper cited above. Look at Table 3, the one that contains results for a no reblance condition.
Rebalancing does win in average return, by an astonishing, massive, .014%. (I hope that had good numerical accuracy control in their simulation.)

Reblancing does somewhat better when you look at volatility. All of the rebalancing conditions have around 10% volatility as versus 12% for a pure buy and hold strategy.

My conclusion: rebalance for less jumping around, not for a better return. If I can get the same return with less volatility why not go for it? If I want better returns, as usual, I'll have to take on more risk.

[ghost9804]

In theory, re-balancing could results in a total higher return than any return from the sub asset class.
I still remember an interesting question:"how do you make money from a random walking stock market, which has zero expected return"
So if you have 2 asset class:
1. cash, stable value, no interest, expected return is zero
2. a stock, random walking, expected return is zero
if you keep the cash and stock to be 50/50 (or any other fixed ratio) and keep rebalancing every time the stock price changes. ignoring any transaction fees, you are guaranteed to make money whenever the stock price come back to it's original price. you can use Excel to simulate and prove it.

the weighted return thing only applies to portfolios without any transactions. Imagine I can catch all up and downs, I can have a return much higher than the reported annualized return of the specific fund. Rebalancing tells you when to catch the up and downs, so theoretically, can results higher returns than the weighted average of all the sub asset class.

[tetractys]

I expect upwards of about 1%. But I don't consider it a bonus; just a return from outlandishly out of balance portfolio assets returning to mean post rebalancing. So far it seems to be working. -- Tet

[Matigas]

"Rebalancing equals noise."

John Bogle

[umfundi]

"Rebalancing equals noise."

John Bogle

Bogle never said that.

He might have said not to worry about small drifts in your equity ratio and that that formulaic rebalancing with precision is not necessary, but so far as I can find he he never made that exact quote.

Keith

[grayfox]

This paper from 2008 that was posted on another thread has the best information I've seen on rebalancing

Opportunistic Rebalancing

A couple of points:

1. Observe that some periods had negative rebal-return and some periods had positive rebal-return. Periods with trending markets had negative rebal-return; periods with reversals, had positive rebal-return. Rebalancing can add to return or subtract from return, depending on the market conditions, i.e. trending or mean-reverting.

2. The amount of rebal-return varied with the rebalancing algorithm. It must also vary with the portfolio asset allocation, as well. The paper only looked at one asset allocation, 60/40 stock/bonds.

3. The often-recommended annual rebalance with 0% bands does not look like the best method. The paper shows 20% bands with look-interval of 1 day to 2 weeks was best.

[livesoft]

...
3. The often-recommended annual rebalance with 0% bands does not look like the best method. The paper shows 20% bands with look-interval of 1 day to 2 weeks was best. ...

... of the limited methods that the paper looked at.

[CaliJim]

[removed by author - didn't read OPs link so my comment wasn't helpful]

[grayfox]

Yes, I believe you have a rebalancing algorithm that is triggered by a large one-day fall in the market. I would categorize that as event driven re-balancing. Not covered by the paper. I guess the theory is that the market over-reacts and is likely to rebound within a few days, so it is a behavioral theory.

But it definitely appears that a short look-interval works best. Check your portfolio once a week to see of it needs re-balancing.

[umfundi]

I think the answer is that bonehead (simply periodic) rebalancing is worth 0.2 - 0.5% a year in a diversified portfolio.

There is a vocal minority that thinks RBD market timing is rebalancing, and they can do much better than a fraction of a percent.

There is another minority that thinks rebalancing will be negative, because they are prepared to tolerate drift in their AA to higher risk.

My own conclusion is that it is worth for me to pay a professional a low fee to pay attention to details like this, and others. He's a passive manager, but I believe his attention outweighs my inattention.

Keith

[BillyG]

This paper from 2008 that was posted on another thread has the best information I've seen on rebalancing

Opportunistic Rebalancing

A couple of points:

1. Observe that some periods had negative rebal-return and some periods had positive rebal-return. Periods with trending markets had negative rebal-return; periods with reversals, had positive rebal-return. Rebalancing can add to return or subtract from return, depending on the market conditions, i.e. trending or mean-reverting.

2. The amount of rebal-return varied with the rebalancing algorithm. It must also vary with the portfolio asset allocation, as well. The paper only looked at one asset allocation, 60/40 stock/bonds.

3. The often-recommended annual rebalance with 0% bands does not look like the best method. The paper shows 20% bands with look-interval of 1 day to 2 weeks was best.

That's a very interesting paper and it makes sense. But I couldn't tell if it is just a backward-looking algorithm derived from the data that has a ring of reason dressed as a halo that makes it look useful.

I wonder if there's an easy way to set up an automatic alert that tells you when a rebalance should be triggered based on hitting a 20% band?

Billy

[grayfox]

Here is what I am thinking:

Without Re-balancing
You want to invest a sum of money for some period. You chose some collection assets for your portfolio. Each asset has an expected return, R_i, and volatility, sigma_i^2. One way or another you select the weights, w_i of each asset. Then you make the investment.

The expected portfolio return R_p is the weighted sum of the asset returns.



The expected volatility of the portfolio, sigma_p^2, is a little more complicated to calculate, because it has to take into account not only the volatility of each asset, but all the covariances between pairs of assets.



At the end of the period, each asset will have some return and volatility. If you held the assets through the whole period, the portfolio return and volatility could be calculated using the same equation above, the weighted sum of the asset returns.

With Re-balancing
Now what happens with rebalancing is that you don't hold through the whole period. You sell some and buy some. You are changing the portfolio partway though the period. The return and volatility of the re-balanced portfolio will be different from the buy and hold. Obviously the difference will depend on the re-balancing algorithm.

Let's call the difference between the buy-and-hold portfolio and a re-balanced portfolio, the rebalancing return, DELTA. R_r = Rp + DELTA

DELTA is random variable. Depending on your portfolio and rebalancing algorithm, DELTA is drawn from probability distribution with some parameters like mean and variance. As I said above, this mean and variance will depend on the rebalancing algorithm.

DELTA can be positive or negative. We've seen that it tends to be negative in trending markets and positive in mean reverting markets.

Re-stating the poll question: For some portfolio and re-balancing algorithm, what is the mean value of DELTA, mean(DELTA)? Is it positive, zero or negative?

Another question is: How big is the variance of DELTA, var(DELTA) ?
Since the return of the rebalanced portfolio R_rp = Rp + DELTA, then variance of the rebalanced portfolio equals the var(R_p) + var(DELTA), if R_p and DELTA are independent. This result is from basic statistics. See Variance of Differences of Random Variables

In other words, re-balancing increases the variance of the portfolio by the variance of DELTA.

[Random Walker]

Gray fox,
Now you're seeing what I was trying to say. Additions and subtractions to the portfolio via rebalancing make it a lot more complicated than just a weighted average return issue. Rebalancing helps in reversion to the mean times and hurts when momentum dominates. Lots of variables . That's probably the best reason to simply use rebalancing for risk control and if there is any extra return, then it's icing on the cake. So returns, correlations, volatility are what matters to the portfolio effects of an asset class. So what do you think of a small CCF component?

Dave