Portfolio Rebalancing Based on Expected Profit and Trading Costs (Redux)

The idea of rebalancing a portfolio to maintain target asset allocation percentages is simple in theory, but tricky in practice. It is not obvious how far asset class percentages should be away from their targets before it makes sense to rebalance. I have recently improved a scheme that I have now fully automated in my portfolio spreadsheet. Instead of obsessing over my portfolio’s returns, a script emails me when I need to rebalance.

Investors should use any new savings or withdrawals they make as opportunities to rebalance by buying low asset classes or selling high ones. However, as a portfolio grows, rebalancing with new savings and withdrawals is unlikely to be enough to maintain balance when asset classes have big swings.

Common advice is to rebalance a portfolio on a fixed schedule, such as yearly. This has the advantage of allowing investors to avoid obsessing over their portfolios all the time, but has the disadvantage of missing potentially profitable opportunities to rebalance. Computing thresholds automatically in a spreadsheet permits me to ignore my portfolio unless my script emails me. This gives me the advantages of threshold rebalancing without the disadvantage of constant monitoring.

When choosing rebalancing thresholds, most experts advise investors to either use percentage thresholds or dollar amount thresholds. For example, you might rebalance whenever you’re off target by more than 5%, or alternatively by more than $2000. However, these approaches don’t work for all portfolio sizes. Percentage thresholds lead to pointless rebalancing in small portfolios, and dollar amount thresholds lead to hourly trading in very large portfolios. We need something between these two approaches.

Computing Thresholds

When asset class A rises relative to asset class B, and then A drops back down again to the original level relative to B, rebalancing produces a profit over just holding. I compute rebalancing thresholds based on the idea that the expected profit from rebalancing should be 20 times the ETF trading costs.

All the mathematical details of how I compute rebalancing thresholds are in the updated paper Portfolio Rebalancing Strategy. I’ll just give the results here.

I have a sub-portfolio with 3 U.S. ETFs. To keep them in balance relative to each other, the spreadsheet starts by computing the following quantities for each ETF:

m – Current portfolio total value times the target allocation percentage. This is the target dollar amount for this ETF.
s – Bid-ask spread divided by the ETF share price.

Other parameters are

c – Trading commission.
f – Desired ratio of trading costs to expected profits. I use 0.05 so that the expected profits from rebalancing are 20 times the trading costs.

The dollar amount threshold for rebalancing then works out to the following formula which may seem a little intimidating, but it only has to go into a spreadsheet once.

t = (m/(2f)) * (s + sqrt(s*s + 8*f*c/m)).

So, it makes sense to rebalance an asset class if its dollar level is below m-t or above m+t. As long as there are at least two asset classes far enough out of balance (with at least one too high and at least one too low), it makes sense to rebalance.

This method works fairly well when the target allocation percentages are close enough to equal. The new part of this work that I completed recently is a more accurate method when there are only two asset classes, but their allocation percentages aren’t necessarily close to equal.

This applies to my case in two ways. I view my stocks as a sub-portfolio with one part denominated in Canadian dollars (30%) and one part denominated in U.S. dollars (70%). The U.S. part contains the 3 U.S. ETFs I mentioned earlier. One level above this, I view my overall portfolio as one part stocks (currently about 80%) and one part bonds (currently about 20%). This bond percentage will rise as I get further into retirement.

The new method for two asset classes looks remarkably similar to the old method. Consider the case of stock/bond rebalancing. Let m be the target dollar amount for stocks and b be the target dollar amount for bonds. Let m’ be the harmonic mean of m and b:

m’ = 2/(1/m + 1/b).

Then the formula for the threshold dollar amount is the same as the earlier formula, except that we replace m with m’:

t = (m’/(2f)) * (s + sqrt(s*s + 8*f*c/m’)).

If the stocks and bonds get further away from their target amounts by more than t, then it’s time to rebalance. The full paper gives further details on how the commission amount c and the bid-ask spread s should be adjusted in cases where extra trading is required, such as when rebalancing involves currency exchange with Norbert’s Gambit.

The difference between this two-asset class method and the earlier method is that the new method gives a lower threshold. The old method would give a very low threshold for the asset class with the smaller target percentage, but a high threshold for the other asset class. But we only rebalance when both thresholds are met. So, the higher threshold dominates. The new more accurate method gives a lower overall threshold, so that we can better take advantage of rebalancing opportunities.

Conclusion

It took me a while to work all this out, but now I don’t have to pay much attention to my portfolio. When I have some money to add, I buy the asset class that my spreadsheet says is furthest below its allocation, and occasionally I get an email telling me to rebalance.

Many readers have asked for a generic version of my portfolio spreadsheet, and I’ve tried to produce one a number of times. But, it’s difficult to make it general enough to be useful. I’m happy to answer questions for those looking to create their own spreadsheets, but I’m unlikely to produce a generic version to work from.