August 27, 2012
Estimating The Optimal Rebalancing Rules
Asset allocation and rebalancing are a powerful team with a history of improving the odds of earning a decent return. But in order to harvest the associated risk premium, you’ll have to deal with two big challenges. One is behavioral—rebalancing works best in a contrarian context, i.e., buy low, sell high. The other hurdle is technical—deciding when to rebalance, and by how much.
Rebalancing is a relatively new research area in finance, but the literature has been expanding rapidly. I devote a chapter to the topic in my book, Dynamic Asset Allocation, although the subject really deserves a book (or two) of its own. In any case, the advice on rebalancing runs the gamut, which means that clarity and simplicity are easily victimized in this corner. But if you’re looking for some relatively unbiased context from a quantitative perspective, a model outlined by Seth Masters at Alliance Bernstein (“Rebalancing: Establishing A Consistent Framework,” Journal of Portfolio Management, Spring 2003 ) is a good place to start.
The basic goal of the Masters model is to estimate the optimal rebalancing trigger points for each asset in a portfolio in an asset allocation framework. In other words, how far should you let a portfolio's asset allocation drift before rebalancing the pieces back to the target allocation? No one really knows the answer, of course, and no amount of analysis can fully solve this mystery, courtesy of our ancient nemesis: an uncertain future. Nonetheless, we can and should focus on developing an approximation of the ideal trigger point by intelligently analyzing the data. We shouldn't be slaves to the result, but going through the process is immensely productive for understanding how rebalancing can help us.
On that score, Masters offers a relatively easy and robust model. The analysis can be run in a simple Excel spreadsheet (or in more sophisticated software packages, such as R). Either way, the strategic insight can be invaluable.
To run the analysis, you'll need estimates for five inputs:
With those numbers in hand, it's a simple matter to plug in the numbers into Masters' formula for estimating the trigger points for each asset in a portfolio:
As a simple of example of how the Masters model works, consider a basic two-asset class portfolio with a target allocation of 60% stocks (S&P 500) and 40% bonds (Barclays Aggregate Bond). Per the variables above, the model requires that we make some assumptions. Let's briefly consider each:
• Investor risk tolerance is hard to pin down, although Masters writes that 5% is a rough estimate for many of his clients and so we'll stick with this number for a test drive.
• Masters also assumes a 1% cost of trading.
• Ideally, the volatility inputs should be projected values, but for simplicity let's use historical data: annualized standard deviations of monthly total returns for the past decade through July 2012 for each asset class.
• Each asset's correlation with the rest of the portfolio (which in this simple example is the other asset), based on monthly data for the trailing 10 years through July 2012.
With the numbers entered, here are the results:
Given our assumptions, the Masters model tells us to rebalance both the equity or bond portion of the portfolio whenever the allocation rises or falls by 4 percentage points.
What can you do with this information? First, a warning: no one should assume that this model gives us the last word on rebalancing rules. There are no silver bullets here, or anywhere else in financial analytics. But it's a valuable starting point—a benchmark for evaluating rebalancing rules for a specific portfolio with a particular set of assumptions. For deeper insight, you should compare the recommendations with advice from another methodology. If the two approaches dispense similar recommendations, that's an encouraging sign that you're in the ballpark for estimating the optimal rebalancing strategy. If they disagree sharply, however, it may be time to rethink your rebalancing strategy, including the underlying assumptions.
Remember, too, that all the standard caveats apply, including the garbage-in-garbage-out risk. The Masters model is only as reliable and robust as the estimates you put into it. But assuming that you can come up with reasonable guesses about the future, the formula can do quite a lot of the heavy lifting for deciding how and when to rebalance a portfolio.
Nonetheless, this model probably works best if you use it routinely and test it by plugging in a range of inputs. In other words, it's important to get a "feel" for how it works and what it's telling us. For instance, if we forecast that the volatility for equities will be 50% higher than the assumption above, that would lower the rebal trigger point for stocks only slightly to 2 percentage points from 3.
Before making real-world portfolio changes, it's crucial to "play" with the model to understand its sensitivity to each input. But don't expect too much. Mere mortals are destined to estimate risk and return with some degree of error. No one's immune. All the more reason to diversify across assets (far more than the two in our simple example), in part to diversify the blowback from error in predictions and less-than-perfect rebalancing decisions. Generally speaking, a prudent set of forecasts for 10 asset classes, aggregated up to the portfolio level, will be more statistically robust compared with predictions for two asset classes. As a foundation, consider holding a mix of the major asset classes via ETFs, index mutual funds, and/or your favorite actively managed products.
Ultimately, the real message in the Masters model is that you should spend a fair amount of time developing risk estimates for each of the asset classes, and the portfolio overall. The good news is that forecasting risk, while far from easy, is somewhat easier than predicting returns directly for asset classes across medium- and long-term horizons. Thanks to Masters, it's also easier to process the estimates for insight on when to rebalance a portfolio. Considering the importance of rebalancing for prudently generating risk premia, that's a big deal, and a potentially rewarding one too.
Correction: An earlier version of this story incorrectly explained Masters formula as it pertains to portfolio correlation and one of the estimates of volatility. The correction focuses on the fact that one of the volatility estimates is related to the rest of the portfolio, i.e., the volatility for assets other than the asset in question. Also, each asset's correlation relates to the rest of the portfolio as well. As such, some of the estimates for the two-asset class portfolio above have been changed. In a portfolio with only two asset classes, the "rest of the portfolio" is, of course, the other asset. Originally, I estimate vol and correlation by looking at the combined portfolio. Correcting this oversight results in dramatically different trigger points in the example. Sorry for the mix-up.
Posted by jp at August 27, 2012 8:28 AM
Yes, you are indeed correct. Thanks! The correlation and volatility related to the portfolio should in fact be the "rest of the portfolio," i.e., separate from the asset class under consideration. Correcting the data in the simple 2-asset class example above results in a much lower trigger point, as you note. I've corrected this in the table and in the definitions stated.
In this example, of course, it's quite easy to estimate the "rest of the portfolio" for vol and correlation, since it's simply the other asset. The calculations become a bit more nuanced in a multi-asset class portfolio.
Meanwhile, the underlying calculations in the spreadsheet are still fine (I just double checked them against the numbers that Masters uses in his article and they match up).
Posted by: JP at August 28, 2012 2:09 PM
interesting article although your trigger points seem, at first glance, extreme
after reading the original article:
j is supposed to be the "rest of the portfolio" (quoting masters paper), so sigma j in your example is just the sigma of the other asset (i.e sigma j = 0.036 for US stocks)
second ro ij is by the same reasonning the correlation with the "rest of the porfolio", which is in this two assets case, the other asset
I just computed the correlation with 10 year T-Bond with the baclays us aggregate (monthly frequency, 10 year period) for a result of -0.05%
all in all, a trigger point of 3.9 percentage points for US Stocks
Am I correct ?
Posted by: rafaminos at August 28, 2012 12:41 PM
It seems that in this simple example, rebalancing would always be triggered by the least volatile asset in the portfolio. The bond would hit it's trigger before the stock. If it hits the high trigger, the funds can go to cash, but if it hits to low trigger, does it force sale of the other assets?
Posted by: Tim at August 28, 2012 12:01 AM