The following question was recently sent to me:** suppose a firm uses a blended benchmark that rebalances quarterly… how should they calculate external dispersion?** I guess I hadn’t given much thought to the possible challenges with rebalancing quarterly benchmarks, but a solution came to me rather quickly, that I want to share with you. Of course, feel free to “chime in,” should you have other thoughts or find some flaw in my logic.

**Rebalancing 101: when to?**

Before we talk about *how to *rebalance a quarterly blended benchmark, let’s briefly discuss the “when to” do it.

In my view, the timing for rebalancing a benchmark should coincide with the timing to rebalance the portfolio. The benchmark is to align with the benchmark’s strategy, yes? And so, one way to do this is to ensure the rebalancing timing aligns.

**How to go about rebalancing quarterly benchmarks**

Let’s say that your benchmark allocation consists of 60% S&P 500 and 40% Bloomberg Barclays’ Agg. You’ve elected to rebalance quarterly. Recall that the **Global Investment Performance Standards (GIPS®)** requires you to provide a 36-month, *ex post*, annualized standard deviation (what is referred to in the question as “external dispersion,” a term I’ve learned to dislike). If you rebalanced monthly, this wouldn’t be a challenge. But since you’re rebalancing quarterly, how do you come up with the appropriate blended benchmarks for the other two (non-rebalanced) months?

We’ll use the following information to walk through the approach I came up with:

We see in the top panel the returns for the two indexes for the three months, as well as our target allocation: 60/40. I’m using notional values to allow us to capture the changes to the benchmark for the second and third months, where rebalancing isn’t occurring.

**Month 1: Standard rebalancing logic**

Month 1 is the month we rebalance, so our Blended Benchmark for the month is simply:

*where: *

*i* represents the different individual indexes

A = the strategic allocation for index i

r = benchmark return for index i

This is pretty standard practice, yes?

**Month 2: Adjusting the blended benchmark to reflect prior period results**

Okay, so now we move to the tricky part: how to blend the second and third months’ benchmark returns without rebalancing? This is where the notional values come in. As can be surmised, we start with $1,000 (or £, €, whatever). We first adjust these values for the effect of the returns from the first month. This is simply:

*where:*

*NV *= notional [dollar] value

*m *= the individual month we’re working on; in this case, it’s the second month

These amounts reflect how each benchmark changed, but using notional values. And these are what we will use to derive our blended benchmark for the second month:

If you follow this same logic for Month 3, you will get the result (4.05%) that appears above.

**Simple, intuitive, logical approach to rebalancing quarterly benchmarks**

I think this is a rather intuitive approach; well, it’s intuitive to me. And, fairly simple (Too simple? Am I missing something?)

What do you think about it? Do you have other ideas, have any issues with my approach, see flaws? Please feel free to send in your comments or questions; thanks!

Oh, and I’ll have a bit more to say on this in the August newsletter. Plus, I’ve decided to incorporate this into our **Fundamentals of Investment Performance Measurement** class! I’ll be teaching an **in-house class** shortly for one of our clients!

You can use weights and compounding rates:

Month 1: Return = w1[1+(month1-index1)Return]+w2[1+(month1index2)Return] – 1

Month 2: Return = w1[1+(month1-index1)Return] [1+(month2-index1)Return]+w2[1+(month1index2)Return] [1+(month2-index2)Return] – 1

Thanks Seddik. I realize it’s been quite a while since you sent your comment in. Our system is designed to alert me when comments arrive, but it doesn’t always; plus, it appears some are simply lost. Oh, well. Thanks for sharing your thoughts.

Stay safe!

Dave

David,

More than intuitive – absolutely the correct way to do it. – Well done.

What you will notice is that the quarterly return that results (1+4.2%)*(1+3.21%)*(1+4.05%)-1=11.90% is equal to the fixed initial weights compounded with the quarterly returns of each asset:

60%*15.75%+40%*6.11%= 11.90% – which sort of proves the calculation!

What you didn’t answer is the calculation of external dispersion (I hate that term as well) – lets call it variability (not volatility which as you mentioned in a blog post recently is not quite the same thing)

Well of course the calculation of variability is exactly as you expect based on the total return of the benchmark – not using any weighting of the components.

Now why rebalance the benchmark quarterly back to 60:40 (if that is what the client wants then OK I guess) – but rebalancing in theory requires transactions (and associated costs) Also if there is any short term momentum in the market the benchmark is always selling the outperforming asset and buying the underperforming asset – also bad (but good if mean reverting)

Bearing in mind if the weights are allowed to float for the entire year then the resulting total return (compounding each month using David’s calculation) will be exactly the annual return applying the initial weights to the annual return. If you don’t believe me try it.

I would argue that in setting a fixed weight the clients are implicitly requiring the weights to constantly move with market movements and should not be rebalanced to the initial weights.

Best regards

Carl

Carl, thank you so much for your comments, encouragement, and additional excellent point: much appreciated!

Carl, I reread your comments, and want to opine on the “why rebalance the portfolio?” Why, indeed, given that there are costs involved?

I suspect this is a highly debated topic. Let’s say that equities are doing great while bonds not so much, and the actual allocation shifts from 60/40 to 80/20 or even 90/10. Why not let this continue forever? Why sell your winners?

Well, in theory (I guess) the client still has a set ratio, which reflects their risk tolerance. When it goes too far away, then they might foresee problems.

I recall after 1999, the year growth stocks had phenomenal years (one of the funds we were in had a return > 100%! We did not rebalance, but let our winnings ride … and they didn’t do too well, as in 2000 the same fund was down around 50 percent (i.e., we gave all of our gain back).

I suspect there are academic papers testifying to the wisdom/foolishness of rebalancing. I, like you, usually just deal with the way it all is.

Thanks, again!

Hi David /Carl, thanks a lot for the contribution on such rare piece of topic. I have found it very intuitive. I did some calc in spreadsheet and thought of contributing to your discussion.

Different clients seeks different requirement. Some look for static target weight and many look for quarterly or annual re-balancing weight.This is strictly client driven choice Benchmark are attached to their specific composite type with their own set of rules/guidelines. But it is also necessary for like vs like comparison for that reason the portfolio and benchmark re-balancing needs to be synchronized, else the game is not fair. Since the blended benchmark has more than one component , its component weights will keep shifting and so will the contribution from each component. It needs to be re-balanced at some point(when weights gets drifted far enough).This would be necessary to be close to the set benchmark target. Also If this does not happen then the risk stats/ratios may also be affected which might be based on benchmark returns.In case if the weights are static then there is no requirement for re balancing. It would simply mean to link the return for three months. However if we follow the actual drifted weight times return then we get a blended benchmark return which can fairly be compared with portfolio return which runs on the same rule.

Month 1 Notional Opening Value Strategic Weight Actual Weight Notional Ending Value Month 1 Return Contribution to Return – Based on actual weight Contribution to Return based on Target fixed weight

S&p 500 600 60 60 630 5 3.0 3

BBAgg 400 40 40 412 3 1.2 1.2

Total 1000 100 100 1042 4.2 4.2 4.2

Month 2 Notional Opening Value Strategic Weight Notional EndingValue Month 2 Return Contribution to Return – Based on actual weight Contribution to Return based on Target fixed weight

S&p 500 630 60 60.46 655 4 2.42 2.4

BBAgg 412 40 39.54 420 2 0.79 0.8

Total 1042 100 100 1075.44 3.21 3.21 3.20

Month 3 Notional Opening Value Strategic Weight Notional EndingValue Month 3 Return Contribution to Return – Based on actual weight Contribution to Return based on Target fixed weight

S&p 500 655.2 60 60.92 694.5 6% 3.66 3.6

BBAgg 420.24 40 39.08 424.4 1% 0.39 0.4

Total 1075.44 100 100 1119.0 4.05% 4.05 4.0

11.46 11.40

Thanks for your contribution to the discussion!

David,

You’re correct ultimately this is the client’s choice – and at some point you will need to review the weights ( but I would suggest less frequently than quarterly – perhaps annually or even less frequently in line with the formal review of liabilities) I take the view that any benchmark that requires frequent transactions is inefficient in practice.

Also don’t forget the point if you let the weights float you get the same return as the initial weights applied to the annual return over the annual period

Best regards

Carl

Carl,

Interesting perspective. An entire lecture (or more) could be built around this topic, no doubt. As always, we appreciate your participation and contributions.

Best wishes,

Dave

Hi David. Good topic that is often misunderstood. Many portfolio managers do the ‘back of a matchbox’ calc by multiplying static weights with monthly returns and expecting it to match a benchmark with quarterly/annual rebalancing. The rebalancing is usually done to ensure that the fund does not breach its mandated limits, else you might end up with conservative fund which has an asset allocation of an aggressive fund. Funds also need to be managed tightly against strategic weights when liability matching comes into play. As mentioned by Carl, the dispersion is independent of rebalancing frequency and purely based on its total return.

Thanks, Pieter. I don’t recall ever seeing a mathematical way to do this, nor do I recall it ever being a topic of discussion. However, when the question arose, I decided to come up with an approach, and the one I’ve documented here (and will elaborate further on in the August newsletter, as well as in our Fundamentals of Performance Measurement course) seemed logical. The idea of simply applying the ratio to the quarterly return would not work, as we need the individual monthly returns for the 36-month standard deviation, though, as Carl pointed out, it can serve as proof that the method does, in fact, work.

Hi David. Good topic that is often misunderstood. Many portfolio managers do the ‘back of a matchbox’ calc by multiplying static weights with monthly returns and expecting it to match a benchmark with quarterly/annual rebalancing. The rebalancing is usually done to ensure that the fund does not breach its mandated limits, else you might end up with conservative fund which has an asset allocation of an aggressive fund. Funds also need to be managed tightly against strategic weights when liability matching comes into play. As mentioned by Carl, the dispersion is independent of rebalancing frequency and purely based on its total return.

Pieter, thanks for your comment … of nearly two years ago! (geez, it’s about time I replied). Our system is supposed to alert me of comments, but apparently doesn’t always, and it appears that some simply get misplaced. Never-the-less, I thank you.

Best wishes,

Dave