## Friday, March 18, 2011

I was sent an attribution problem by a client to research for them. The details appear here:

I've highlighted the issues they were concerned with. First, in yellow we find that the technology sector's allocation effect is positive (0.222%), even though (a) it's overweighted (75% vs. 67%) and (b) the sector's index return (7.37%) is below the overall index return (7.51%).  How can this be? Note that the Brinson-Fachler model was used, meaning that this situation should result in a negative allocation effect. Why? Because the manager overweighted a sector that underperformed the overall index return (meaning there were other sectors that performed better).

Second, in blue we find that the finance sector's selection effect is positive (0.212%) although the portfolio underperformed the index (7.47% vs. 7.79%). The model expects the effect to be negative in this case. And why? Because the manager underperformed the index, so the selection effect should be negative to reflect poor selection decisions.

And so, what's the problem?

Well, if you duplicate these numbers you'll see that the sector effects were arrived at by adding the effects of the underlying sub-sectors. This isn't the way it's done. We need to apply the attribution effect formulas to the sector levels. And when we do that, we get:

By calculating the sector effects using the same formulas as we use at the subsector level, our numbers make sense. The technology sector's allocation effect (-0.011%), as well as the finance sector's selection effect (-0.105%), are now both negative, as we'd expect.

Conclusion: sometimes the numbers don't add up ... because they're not supposed to!

1. Dave:

You are absolutely correct in saying that you CANNOT simply run the Brinson model over the SUB-SECTORS relative to the total portfolio and then add the results. This is a common mistake made by a great many performance analysts. As you point out, you must approach this by first analyzing the impact of the SECTOR weightings and returns relative to the total index. And, your sector allocation results are exactly right.

However, this is where your analysis breaks down. You must next examine the impact of sub-sector weightings and returns in the context of their own sectors, and then pro-rate these results back to the entire portfolio. When you do this, you will have two valuable results. First, you will have analyzed each sector in its own right, answering the question: "Why did MY sector outperform the index sector?" And you will explain this by looking at your relative sub-sector weights and returns. The results will tie out exactly to the absolute difference in return between the portfolio sector and the index sector. Second, you will answer the equally relevant question: "How did each sector and sub-sector contribute to the excess return of the overall portfolio?" To solve for this, you simply pro-rate the individual sector attribution results by the weighting of each sector.

I had presented this hierarchical process in my article "Balanced Portfolio Attribution" which was published in the Journal. The methodology is not new; I believe there are several approaches which are similar if not identical and which may go by different names ("nested attribution" or "decision based attribution" etc.) In all cases, the goal is to identify the 3 decisions at work: a) to allocate actively to broad sectors, and b) to allocate actively to sub-sectors or industries and c) to manage the specific investments within each sub-sector.

When you approach the attribution analysis this way, you find that the results DO add up. The slight confusion caused by the Finance sector's positive selection effect (which is correct) was caused by the blurring together of sub-sector allocation and selection effect in your analysis. A hierarchical approach shows that the sub-sector allocation effect was quite negative, underweighting the best sub-sector (Banking) and overweighting the worst sub-sector (Insurance.) If you look at the relative absolute returns of these sub-sectors, you find that 2 out of 3 of them outperformed the benchmark. So, once you strip out the active sub-sector allocation effects, you do find a positive selection impact.

The results of a hierarchical approach are:

Sector allocation: - 3 bps

Subsector allocation: +9 bps

Selection: +34 bps

Total: +40 bps

So, the results DO add up. But let's remember Andre Mirabelli's wise advice that "just because the numbers add up, it doesn't mean the analysis is correct." In this case, a hierarchical process IS correct because it reflects the active decision process of the investment manager.

2. Thank you for your sharing this. I will do this analysis, too.