## Not for multi-periods, but for single periods, geometric attribution has residuals!

In a recent post about multi-period attribution, I mentioned that while it’s true that arithmetic attribution is “linking challenged,” resulting in residuals that require a smoothing factor so that the linked excess returns match the linked excess return, geometric attribution has residuals for single periods (perhaps we might call it “singe period challenged”). I thought I should explain in some detail why I state this.

## Proof…comparing the formulas, mapping the factors

I’ll share with you three slides from our attribution course, which I’m in the process of revising. In the course, we discuss geometric attribution, and review the model that Carl Bacon developed: it’s a geometric equivalent of the Brinson-Fachler model.

Recall that with arithmetic models, we reconcile to the arithmetic version of excess return (simply, portfolio return (RP) minus benchmark return (RB)). Geometric reconciles to the geometric version ((RP+1)/(RB+1)-1). Returns in the models must be converted to their geometric equivalent.

Let’s first consider how we would extend the arithmetic version of Brinson-Fachler to geometric (click on the slide to enlarge it):

What I show is the conversion of the return differences from arithmetic to their geometric equivalent. As Carl explained to me, when he was developing his model the numbers didn’t work out. And so, he needed to add another factor.

## Introducing, the semi-notional return

Carl introduces what he calls a “semi-notional” return (i.e., half notional, or half benchmark, where notional = benchmark):

Mathematically, it’s the “sum product” of the sectors’ portfolio weights and benchmark returns. Carl uses this return as part of his adjustment factor:

As you can see, this corrective factor is the sector’s benchmark return, plus one, divided by the portfolio’s semi-notional return, plus one.

The following shows the mapping of the attribution factors from arithmetic to geometric:

We see the use of a factor that has **no corresponding expression** in the arithmetic version of Brinson-Fachler. Without it, the math simply won’t work; i.e., **we’ll get residuals**.

## Myth #2: geometric attribution doesn’t have an interaction effect

While we’re at it, let’s tackle another myth: that geometric attribution doesn’t have an interaction effect. It won’t, if, like Carl, you use the portfolio weight in the selection effect; however, if we use the benchmark weight …

If, like me, you see value in reporting the interaction effect, simply replace the portfolio weight with benchmark weight in the selection effect, and introduce the geometric version of geometric attribution … but don’t forget to include Carl’s adjustment factor!

I want to point out that on this topic, Carl is in agreement with me: geometric can have an interaction effect. He sees little (if any) benefit to interaction, which is fine; just another topic we can (and have) debate. The point here is to dispel the belief that geometric is interaction-effect-free … it isn’t!

## Confession time

I taught Carl’s method as part of our attribution class for several years, but it hadn’t dawned on me that this was, in fact, an adjustment (smoothing, corrective) factor, until I reached out to Carl a few years ago, inquiring into what it was there for. That’s when he explained that it was necessary for the numbers to work. It still didn’t quite “sink in” until I reread Jose Menchero’s article on geometric attribution several months ago, and came to his explanation that geometric needs a factor to make the math work.

## Summary

I believe I’ve demonstrated that, like arithmetic, geometric attribution has residuals: the only difference is that arithmetic needs it ** across time periods**, while geometric requires it

**. In addition, I’ve shown how we can have an interaction effect with the geometric approach.**

*within*time periodsWhile I understand that many firms prefer the geometric method, it, like arithmetic, has some challenges which, like arithmetic, can be overcome.

As always, your thoughts are welcome!

David,

Brinson terms were all originally derived as single period decompositions for a single period. I believe it is a mistake to take these single period results as the starting point for a multi-period attribution analysis. And I also believe that it is a mistake to view attribution as a decomposition. (7 = 4+3 = 5+2 = 6+4-3, but ‘So what?’)

As you point out, doing these sometimes retains their meaningless incorporation of the Interaction effect and also leads to the warping (smoothing) of results in order to make the single period terms fit into a multi-period analysis.

When the multi-period decision analysis for either arithmetic or geometric attribution is addressed directly, no interaction terms show up, and often no smoothing is required to get a precisely correct analysis of multi-period decision-type results for each day that roll up to the fund-level results.

Such Attribution values are order-dependent. We have a natural order for time periods. Within a day, we have the prescribed order for decisions at different levels. However, we do not have an ordering of decisions at the same level. For example we usually do not know if the Industry Allocation decision within one sector came before or after the Industry Allocation decision within a different sector.

Thus, for geometric attribution, no smoothing is required at the level of decision types (such as total Industry Allocation), but is required when one wants to assess, for example, the individual Industry Allocation attributes described above.

Due to the math of Arithmetic Attribution, this problem of ordering within a period does not arise. But this is never due to forcing the math to work. Rather it is always due to correctly modeling the economic meaning of each term and then simply noting that they roll up to the appropriate higher level (such as Fund) results.

Thus, it is neither an arbitrary decision nor a bow to the tradition began by Brinson to use fund or portfolio weights in certain formulas. One needs to use the weights that address the intended question addressed by the term, not any old weight or other that can be made to roll up to the fund level.

What is important is that one addresses meaningful questions and not approach Attribution as if all one needs to obtain is some decomposition or other.

Directly addressing the economic intent of attribution leads to the models I built for Opturo. As one might expect, doing this produces single period, two step decision-type results that match various versions of the original Brinson Performance Attribution results. But their generalization to more complicated decision trees, to multiple periods and to risk attribution does not lead to economically meaningless interaction terms or to smoothing whenever orderings exist.

Andre

Hi Andre

Do you mind sharing your attribution model, including a simple example for both single and multi-period? It would be interesting to see how you managed to build a model without residuals or smoothing, and compare it to the Brinson models. Is your model only used by Optura, or by other software vendors as well?

Regards

Pieter

If one rescales the Brinson factors so that they add up to the geometric version of excess return, the asset allocation factor that results is equivalent to the Bacon version presented above. This rescaling is Step 1 in my simple-minded decomposition of multi-period geometric excess return that turns out to be equivalent to the Frongello method—see volume 18 no. 2 of the Journal of Performance Measurement. This link to Frongello justifies the asset allocation term in my mind.

If selection and interaction were treated the same way, and combined into one selection factor as Carl likes to do, the resulting asset allocation and selection terms would add up to the geometric excess return. But the Bacon terms do not add up they compound, so the Bacon selection term remains different and, to me, mysterious.

Andre,

Couldn’t have put it better myself.

David,

For the avoidance of doubt there are no residuals in single period geometric attribution -it is not challenged – that is the point. The selection formula you show above is derived directly from the total selection effect (1+RP)/((1+ RS) -1 which is exactly equivalent to the the total selection effect in the original Brinson model RP-RS . Try it yourself the total portfolio return minus the semi-notional fund gives the total selection effect. .

Regards

Carl

Tim,

I must admit when I initially derived the selection term it was a bit mysterious to me too. It took a while to work out the economic rationale. I hate the term adjustment (because there is no adjustment in truth – it is just what it is) but for want of a better word the adjustment term shown above is necessary because equivalent geometric outperformance in each market does not make the same contribution of excess return to the overall portfolio. Geometrically outperforming a strongly performing market will add more value in dollars and cents than the same level of outperformance in a weakly performing market.

By the way the allocation effects sum arithmetically and the selection effects sum arithmetically – the sums both compound to generate the total geometric excess. Because the geometric excess returns compound through time it means the sums of selection and allocation compound across time as well (but not the individual components) .

Remember the justification to use geometric excess return is not that the geometric attribution method is better (although there are multi-period benefits) its simply that the geometric excess return is more appropriate to use and the arithmetic excess return is flawed. That’s a whole different discussion I won’t get into right now

Best regards

Carl