Sector Attribution with Zero-Weighted Sectors

This page deals with the issue of how to calculate sector attribution with sectors that have a weight of zero either in the portfolio or the benchmark.  The reason why zero-weighted sectors can create problems in sector attribution is that a sector with zero portfolio weight never has a portfolio sector return, and a sector with zero benchmark weight rarely has a benchmark sector return.  This missing returns make it possible to just plug numbers into the attribution formulas, unless of course one simply invents a value for the missing return.  We will show that inventing a value for the missing return is completely unjustifiable, since it produces results that are not amenable to rational explanation.

Problem 1

As an example of the kind of problem where the portfolio sector return is missing, see the following table:

The difficulty here concerns the Small Cap sector.  The portfolio had no assets invested in this sector.  Hence, it was not possible to calculate a return for the portfolio's holding in this sector.  This is a crucial point.  Sometimes, when people are doing the calculations in a spreadsheet, they leave this cell blank.  This is equivalent to supplying zero as the portfolio sector return.

If one uses (either explicitly or implicitly) zero as the portfolio sector return for Small Caps, this gives the impression that the Small Cap sector held investments during the analysis period, and that these investments substantially underperformed the sector benchmark return of 4.0%.  In turn, this will lead to the following attribution results:

Aside from the fact that this table shows a 0.0% return for the Small Cap sector (when in fact there simply was no return), there are two other problems with the numbers in this table:

1. The -0.80% stock selection for Small Caps gives the impression that somebody made poor stock selection decisions in the Small Cap sector.  However, the reality is that there were no stock selection decisions to be made in the Small Cap sector, since there were no assets invested there.  This result also makes the overall stock selection result for the portfolio appear to be zero, whereas in fact, the sector managers in Large Caps and Mid Caps (who were the only managers in charge of invested assets) actually did an excellent job of selecting securities.
2. The 0.80% interaction result is large relative to the other attribution results, and it is not susceptible to any clear or intuitive explanation.  This stands in contrast with the interaction numbers for the Large Cap and Mid Cap sectors, which are smaller in magnitude, as well as being very intuitive (because in both cases, they indicate that the asset allocator overweighted sectors which outperformed their sector benchmarks).

In short, there are many reasons why it is not a good idea to simply "plug in" a portfolio return of zero for the Small Cap sector, then proceed as normal with the attribution calculations.

Proposed Solution to Problem 1

The proposed solution to problem 1 is that sectors for which there is no portfolio sector return should be treated as a special case.  Under this special case, the Asset Allocation Attribute is calculated normally, and Stock Selection and Interaction are defined to be zero.  Also, as mentioned earlier, it is essential that the portfolio sector return be depicted as "N/A", or some other symbol that clearly indicates the absence of a return.  The following table shows the results that this produces for the example we have been considering:

You can see from this example that the asset allocation number does in fact explain the total value added for the Small Cap sector.  Appendix 1 of the journal article (to appear soon), proves mathematically that this holds true in general.

Because there were no assets invested in the Small Cap sector, it seems abundantly clear that it would have been impossible to add or subtract any value by stock selection in that sector.  Hence, it follows that the stock selection attribute for Small Cap should be zero.  In turn, because the stock selection and interaction attributes always sum to zero when the portfolio sector weight is zero (See Appendix One of the forthcoming journal article), it follows that  the Interaction attribute for Small Caps should be zero.

These overall results seem much more intuitive than the earlier results we obtained by "plugging and chugging" the value zero into the portfolio sector return for Small Caps.

Problem 2B

The second problem concerns a portfolio that has invested outside its benchmark.  In this problem, one could perhaps consider the Small Cap sector to represent stocks whose market capitalization is so small that they are outside the benchmark index.  This means that the benchmark weight for Small Caps is zero.  It also means that Small Caps are unlikely to have a portfolio sector return.

The following table shows the basic inputs for this problem.  It also shows the attribution results, based on the very questionable assumption that one can simply "plug in" a benchmark return of zero for Small Caps, then proceed as normal with the calculation.

The shaded cells indicate values that appear questionable.

One important key to this problem is whether it is reasonable to assume a zero sector benchmark return for Small Caps.  We need to be very clear about this.  There will be cases where a sector has a benchmark weight of zero, yet there will still be a very clear benchmark index for the sector.  For example, suppose that the overall benchmark return for the portfolio was "Index excluding Small Caps": the Large Cap and Mid Cap sectors would have non-zero weights in that benchmark, yet the Small Caps (being outside the benchmark) would have a benchmark weight of zero.  One could still use the Small Cap benchmark return in the attribution calculations under those circumstances, and if that return happened to be zero, so be it.  This would be what the journal article calls "Problem 2A".  There is nothing difficult or particularly unusual about that Problem, since the benchmark sector return exists.  It can be treated absolutely normally.

The more interesting case, which we are examining here, is when the benchmark sector return is not known.  The journal article designates this as Problem 2B.  If the benchmark sector return is not known, it is quite wrong to simply "plug in" a value of zero then proceed with the calculations.  The above table illustrates some of the problems that come out of this:

1. If the benchmark sector return for Small Caps was zero, this would seem to indicate that the 5.0% portfolio sector return was the result of excellent management in that sector.  However, the zero benchmark return has simply been snatched out of the air, and may not be truly representative of the investment opportunities that were available in Small Caps.

2. The Asset Allocation result that flows from assuming a zero sector benchmark return is -50 basis points, which seems to indicate that the asset allocator made a poor decision by putting 20% of the portfolio into Small Caps, even though Small Caps were not part of the benchmark.  This is quite wrong, since in fact the Small Cap sector performed better than any other sector in the portfolio, and the best way for the asset allocator to have improved the portfolio performance would have been by putting as much as possible into Small Caps.

3. The large Interaction number that flows from assuming a zero benchmark sector return is 100 basis points.  It is not at all clear what this means.  This attribute is very large compared with the other attributes.

Once again, we see that simply plugging zero into the usual equations is not a very good solution to the problem of a missing input.  It can lead to results that are unintuitive or just plain wrong.

As a better solution, what we suggest is to assign all the value added for the Small Cap sector to the Asset Allocation attribute (this means that Stock Selection and Interaction will be zero).  Additionally, since there is no benchmark return for Small Caps, it is better to display this as "N/A" (or similar), rather than simply plugging in a guessed value such as zero.  These results are shown in the following table:

This result makes sense.  Because there is no benchmark return for the Small Cap sector, it is impossible to assign any particular value to the Stock Selection attribute.  The asset allocator is given full credit for having put assets into a sector that performed well.

Note that, in a sense, the recommended solution is a "second-best" option.  Clearly, if it is possible to discover a meaningful sector benchmark return for Small Caps, this will make it possible to determine how much value was added through Stock Selection, and how much by Asset Allocation.  Therefore, even though the benchmark weight for Small Caps is zero, it would make sense to explicitly identify a benchmark index for this sector.

In the absence of a clear benchmark return for Small Caps, the best that one can do is to give the asset allocator credit for all the value-added arising from the Small Cap sector.

Calculating the Total Value-Added for a Sector

In both of the problems shown above, the recommended solution is to assign all the value-added for the zero-weighted sector to the Asset Allocation attribute, while the other attributes for the zero-weighted sector will be zero.  This raises the question of how one can calculate the total value added for a sector.

One could calculate the attributes using a bogus zero value in place of the missing portfolio or benchmark return.  In both the examples shown above, this does indeed lead to a set of attributes whose total is correct.  However, calculating the total this way would be a very desperate remedy, since it would reinforce in people's minds the legitimacy of "plugging and chugging" a zero value in place of a non-existent return.  This "plug and chug" approach has led to many misunderstandings about performance attribution, so it seems desirable to find a better and more direct formula for the total value-added by any sector.

In both the problems shown above, there is only one zero-weighted sector.  Since it is possible to calculate the total value-added (as the simple difference between total portfolio return and total benchmark return), and it is also possible to calculate the attributes for the other sectors, one can then infer what the total value added for a single zero-weighted sector should be: It will be the difference between the total active return and the sum of the attributes for the non-zero-weighted sectors).  However, if there was more than a single zero-weighted sector, this method would not be available.  In any event, what we seek is an elegant and direct method for calculating the total value-added for any sector in a sector attribution, even if it is zero-weighted (and hence missing a portfolio or benchmark sector return).

To derive the formula that we need, we start with the usual formulas for sector-level attributes in a sector attribution, and we sum them.  Then with a little bit of grouping and rearrangement, we finish with a useful formula for the total value added by a sector:

The last line in this derivation is the formula that we want.  It enables us to calculate the total value added for a sector, even in the difficult cases of zero-weighted sectors.  It consist of three terms:

1. The first term is the sector's contribution to the total portfolio return;

2. The second term is the sector's contribution to the total benchmark return; and

3. The third term is the sector's active weight, multiplied by the total benchmark return.

The first and second terms are the one where missing data could cause problems in zero weighted sectors.  For example, for a portfolio with zero benchmark weight, the benchmark sector return may be missing.  However, in this case, the missing benchmark return would be multiplied by a zero benchmark weight, and so that term of the formula would simply evaluate to zero. In this way, the formula is actually very versatile, because it can be evaluated under any circumstances.

Once you have detected that there is a missing return due to a zero-weighted sector, you can quickly evaluate this formula, and assign the result to asset allocation (with zero for stock selection and interaction).

Doing the Calculations Over Multiple Periods

Over the course of time, any particular sector may enter or leave the portfolio or benchmark from time to time.  For example, a portfolio might start with a benchmark of 50% bonds, 50% equities, but then the benchmark may change to 50% bonds, 40% equities, 10% international bonds.  In a similar fashion, the portfolio managers might choose to reduce their holding of international bonds to zero, and then they might buy back into this asset class after a period of time.  Consequently, any robust performance attribution system must be able to deal smoothly with sectors moving into and out of the portfolio and the benchmark.

As the following matrix (Figure 1) summarizes, there are four possible states of affairs based upon the criteria of whether a sector is in or out of the portfolio or benchmark:

• A) the sector has a zero weight in the portfolio and the benchmark.  Perhaps it sounds trivially obvious, but an attribution needs to know that the attributes will all be zero in this case;

• B) the sector has a zero weight in the portfolio, but a non-zero weight in the benchmark.  This is described above in Problem 1.  The recommended solution is for all the value added to go into asset allocation.  This solution is reasonably clear-cut.

• C) the sector has a non-zero weight in the portfolio, but a zero weight in the benchmark.  This is described above in Problem 2B.  The recommended solution (again) is for all the value added to go into asset allocation.  However, if one supplies a benchmark return for the sector (even though its weight is zero), it would then be possible to calculate the attributes normally.  After all, even though zero weights are an obvious symptom of the problem, the difficulty in calculating attributes arises from the absence of a sector return.

• D) the sector has non-zero weights in both portfolio and benchmark.  This is the only case that is considered in conventional treatments of the topic.  In this case, the usual formulas (as described on the Sector Attribution page) apply, and there in no particular difficulty in doing the calculations.

In general, when an attribution system is doing the calculations over periods of months or years, it is unlikely that every single sub-period (be it a month or a day) will fall into category D.  In each single period, the system just needs to apply the rules shown in Figure 1.  Then, the single-period attributes can be combined over the entire analysis period using whatever multiperiod method one prefers (see the Multiperiod Attribution page for more details).

The following table shows a specific example of how to apply these rules: