Sector Attribution

This page describes sector attribution.  This is also known as "Brinson attribution" or "top-down attribution".

Terminology

Before proceeding, it seems sensible to establish clearly the meaning of certain key terms.

For any given analysis period (e.g. one day, one month), the benchmark will achieve some particular rate of return.  The portfolio will also achieve some particular return.  The difference between the portfolio return and the benchmark return is the active return.  The aim of attribution analysis is:

• to identify which portfolio management decisions contributed to the active return; and
• to assign a meaningful component of the total active return to each management decision.

In this way, we get an explanation of what happened in the portfolio to make it perform differently from the benchmark.

The sector attribution model assumes that there are two different kinds of decision that portfolio managers can make to generate active return:

1.  The portfolio manager can hold any sector over- or under-weight relative to the benchmark.  This kind of decision is known as Tactical Asset Allocation (or simply asset allocation).  For example, if the benchmark weight for global equities is 30%, but the portfolio manager (PM) is particularly bullish about global equities, the PM may choose to allocate 40% of the portfolio into global equities, in the hope of achieving higher returns.

2.  The portfolio can add value within a sector, principally by selecting stocks that perform well.  For example, a global equities manager might choose within the automobile manufacturing sector to overweight Toyota and underweight General Motors.  This decision is known as stock selection.

A lot of the flexibility in the sector attribution model flows from the fact that you can use virtually any sector scheme you can think of.  For example, you could do attribution by country, or by industry sector, by region, or even against a sector scheme where the stocks have been ranked into deciles according to their P/E ratio.

First Principles

The first principles of the sector attribution model are this diagram, and the four equations that follow it:

• Asset Allocation = II - I
• Stock Selection = III - I
• Interaction = IV - III- II + I
• Total Value Added = IV - I

Note: These first principles form the basis for the analysis done in the several papers by Brinson et al.

Let us first examine the diagram.  It shows that, if we combine benchmark or portfolio sector weights (from the left axis) with portfolio or sector benchmark returns (from the top axis), we can obtain returns for four different notional portfolios.  The first notional portfolio (I) is in fact the benchmark.  The second notional portfolio (II), combining portfolio sector weights with index sector returns, captures what would have happened if the asset allocation was active, but the stock selection was strictly in line with the benchmark.  So it is called the Active Asset Allocation Fund.  The third notional portfolio (III) combines benchmark sector weights with portfolio sector returns.  This captures a situation where the asset allocation is strictly neutral, but the stock selection is active.  Predictably enough, it is called the Active Stock Selection Fund.  The fourth and final notional portfolio (IV) combines portfolio sector weights with portfolio sector returns: and this is indeed the actual portfolio to be analyzed.

So, to summarize, quadrant I contains the benchmark return, quadrant IV contains the portfolio return, and quadrants II and III contain the returns for notional portfolios that reflect "pure" asset allocation or stock selection (respectively), with everything else held neutral.

At the risk of stating the obvious, the numbers in quadrants I through IV are returns, and hence they are compoundable.  This fact becomes useful when we consider the problem of multiperiod attribution.

Turning now to the equations, it is easy to see how to obtain the total-level management effects.  The value added by asset allocation is simply the extent to which the notional asset allocation portfolio's return exceeds the benchmark return (i.e. II - I).  Similarly, the value added by stock selection is the extent to which the notional stock selection portfolio's return exceeds the benchmark return (i.e. III - I).  Axiomatically, the total value added is the portfolio return less the benchmark return (i.e. IV - I).  The asset allocation and stock selection effects do not sum to the total value added: the difference is covered by the interaction term, whose formula may not be intuitively obvious (i.e. IV - III - II + I).  However, it is at least obvious that when one sums asset allocation, stock selection, and interaction, one obtains the expression for total value added:

Asset Allocation + Stock Selection + Interaction = (II - I) + (III - I) + (IV - III - II + I) = IV - I

This is the basic framework for performance attribution as presented by Brinson et al.  There are several directions in which one can extend it, but it is very worthwhile understanding these first principles before proceeding to the extensions.

Worked Example:  This is the example from sheet one of the spreadsheet Toward consensus on multiperiod attribution.xls.

The portfolio contains only two sectors (equities and bonds).  The following table shows the weights and returns for these sectors:

How can we work out the total value added by asset allocation, stock selection, and interaction?  Easy -- the first step is calculating the notional portfolio returns, as shown in the "first principles" diagram:

I (Benchmark) = 50% x 2% + 50% x 4% = 3.00%

II (Active Asset Allocation) = 40% x 2% + 60% x 4% = 3.20%

III (Active Stock Selection) = 50% x 4% + 50% x 5% = 4.50%

IV (Portfolio) = 40% x 4% + 60% x 5% = 4.60%

We can then move on to calculate the total-level attributes using the simple equations shown in the "first principles" section.

Asset Allocation = II - I = 3.20% - 3.00% = 0.20%

Stock Selection =III - I = 4.50% - 3.00% = 1.50%

Interaction = IV - III- II + I = 4.60% - 4.50% - 3.20% + 3.00% = -0.10%

Total Value Added = IV - I = 4.60% - 3.00% = 1.60%

These results are intuitively satisfying:

• The +20 basis points of value added for asset allocation came from the tilt away from equities (whose benchmark return was 2%) toward bonds (whose benchmark return was 4%);
• the +150 basis points of value added for stock selection came because equities outperformed its benchmark by 2%, and bonds outperformed its benchmark by 1%; and
• the 10 basis points lost through interaction are due to the fact that the asset allocation tilt was toward bonds, where the stock selection was not as superior as in the equities sector.

As expected, these results all seem consistent and plausible.  The process is so simple, also.  But this example is important because it reinforces our understanding of the first principles of sector attribution.

Before going forward to more challenging territory, let's review the history of sector attribution.

History

Sector attribution is often known as the Brinson method.  This is due to Brinson and Fachler (1985), Brinson, Hood, and Beebower (1986), and Brinson, Singer, and Beebower (1991).  The central argument of all these papers was that the vast majority of a typical portfolio's returns are explained by its strategic asset allocation (rather than tactical asset allocation or stock selection).  Brinson et al. summarized the results of their 1991 study, "While active asset allocation contributed a net underperformance of 26 basis points, and security selection contributed a gain of 26 basis points, neither figure is statistically different from zero". (p. 44)

For example, if an investor chooses to put their assets in a domestic equities portfolio instead of a global bonds portfolio, this choice of asset class will (other things being equal) do more than anything else to determine the return that they receive.  Even though there are differences in the performance of different domestic equities portfolios owing to stock selection or tactical asset allocation, these differences tend to be much smaller than the differences between domestic equities and other asset classes such as domestic or international bonds.

The papers by Brinson et al. were principally concerned with the properties of funds in general, rather than with the analysis of individual funds.  For example, Brinson, Singer, and Beebower (1991) examined ten years of quarterly performance data for 82 pension funds.  They found that benchmark returns accounted, on average, for 91.5% of the variance in portfolio returns [Brinson et al. 1991, p. 45].  This would suggest that choosing the benchmark will have far more influence over a fund's returns than any stock selection or tactical asset allocation decisions made by the portfolio manager.

In this sense, Brinson et al. were making a case that, in general terms, stock selection and tactical asset allocation are less important than people often think.  Contrast this with the approach that a performance analyst takes when calculating attribution on a portfolio.  For the performance analyst, performance generated by the benchmark is completely irrelevant, and the key focus is on the way that sock selection and tactical asset allocation decisions have generated the active return.

Beyond any doubt, Brinson et al. used exactly the same conceptual framework that lies at the heart of sector attribution as commonly used these days.  But one point is too easy to overlook: Brinson et al. were not principally concerned with the problem of how to do attribution on a single fund (e.g. by calculating attributes at the sector level, or by combining sector-level attributes over time).  Rather, they were principally interested in showing that the choice of benchmark, rather than the active management effects, explained most of the variance in portfolio returns.

The importance of understanding this historical background is that some texts about attribution make it seem that Brinson et al. specified two distinct methods for calculating  performance attribution on individual portfolios, when in fact they did not.  Crucially, the paper by Brinson, Hood, and Beebower (1986) did not touch on the subject of how to calculate sector-level attributes.  The only formulas they provided used sigma notation; i.e. they dealt only with the total-level attributes.

Thirteen years before Brinson et al. described the idea of notional portfolios, the Society of Investment Analysts (1972) published their monograph, The measurement of Portfolio Performance for Pension Funds.  They only needed to use three notional portfolios, since they (implicitly) included the interaction term with the stock selection term.  For example, their description of how to calculate stock selection was by taking the difference of two notional portfolio returns:

• "Actual Fund (actual investments, actual proportions) minus
• Partly Restrained Fund (index investment, actual proportions)" [p. 9].

It seems certain that the concept of using notional portfolios to attribute portfolio returns was known to many people by 1974, when the Society of Investment Analysts published a revised version of their monograph.  The exact detail of how these ideas developed seems to be lost in the mists of time.

Deriving the Familiar Sector-Level Equations

For many people, the familiar equations for calculating asset allocation, stock selection, and interaction, at the sector level constitute the conceptual core of sector attribution.  However, we like to encourage the view that the first principles of sector attribution (as described above) are the most important things to understand.  The sector-level equations can be derived from first principles, as we now show:

The first principles show us that the value added by stock selection is the extent to which the active stock selection fund outperforms the benchmark.  Putting it mathematically (see the bottom of this page for a guide to notation):

The inner term from this sum can be interpreted as the stock selection value added for the i'th sector.  This result will be summarized below.  The intuition behind it is that the value added attributable to stock selection is proportional to the portfolio's outperformance of its benchmark return, weighted by the benchmark sector weight.  By assuming a benchmark sector weight, this calculation assumes a neutral asset allocation position.  In this way, it evaluates the stock selection effect independently from asset allocation effects.  Putting it simply, this equation answers the question, "How much value would the stock selectors for this sector have added to the portfolio return if the asset allocation was neutral, but the stock selection was active?"

For the interaction term, we can also easily derive a weighted-sum formula whose inner term is an intuitive and appropriate way to measure the interaction attributable to a single sector:

Once again, there is a clear intuition behind the inner term of this equation.  The interaction term for a sector is equal to the over/under weight for the sector (which is decided by the asset allocator), and the sector's outperformance of its benchmark return (which is decided by the stock selector).  In this way, one can see how interaction is due to the combined operation of asset allocation and stock selection.

Finally, we come to the asset Allocation term.  Working from first principles, we quickly obtain a useful-looking formula:

This formula for asset  allocation has been labeled the "Brinson Hood Beebower" (BHB) approach (see, for example Spaulding, D. Investment Performance Attribution, McGraw Hill 2003, p. 33 and Campisi, S., "Debunking the Interaction Myth", Journal of Performance Measurement, Summer 2004, pp. 63-70).  This is where one needs to take great care in interpreting the algebra.  Since this formula follows validly from first principles, there is no denying that it accurately captures the total value added by asset allocation in accordance with the first principles set out above.  However, this doesn't imply that the inner term of this weighted sum provides a useful measure of asset allocation for each sector.  We discuss this issue further under the heading Variations on this Approach: Mistaken or Beneficial? (see below).

To obtain a different formula, one can effectively add zero to the right hand side of the above formula.  Clearly, adding zero doesn't change the sum, but it does make it possible to find an expression of the weighted sum where the inner term has a more intuitive interpretation at the sector level.  As we will see, it proves useful to take advantage of the axiom that the sum of benchmark sector weights and the sum of portfolio sector weights are both exactly 1, and consequently the difference between these sums is exactly zero:

Continuing the derivation of a sector-level formula for asset allocation, we add zero to the right hand side of the formula we have already obtained:

The inner term from this formula provides a familiar method for calculating the value added attributable to asset allocation for any given sector.  Essentially, the asset allocation attribute multiplies two factors:

1. The amount by which the sector's portfolio weight exceeds it benchmark weight; and
2. the amount by which the portfolio's benchmark return exceeds the benchmark return for the overall portfolio.

The second of these points merits some reflection.  It seems reasonable to say that the asset allocator's job entails identifying the sectors whose benchmark returns will be low or high.  The asset allocator then has the task of underweighting sectors whose benchmark return will be lower than the overall benchmark return, and overweighting sectors whose benchmark return will exceed the overall benchmark return.

Summary of Formulas for Calculating Sector-Level Attributes
	[ Equation 1 ]
	[ Equation 2 ]
	[ Equation 3 ]

Three Attributes or Two?

We have already seen that the Society of Investment Analysts didn't use an interaction term in their analysis.  Like them, many people have, over the years, preferred to effectively combine the interaction term with the stock selection term.

Algebraically, it is simple to combine the stock selection and interaction terms.  As one can easily observe, if one adds these terms, as expressed in Equation 2 and Equation 3 above, the result is simply:

The simplicity of this combined expression can reinforce the idea that interaction is an inconvenient and unnecessary construct, which can be eliminated by the simplest algebraic manipulation.  However, as the discussion on the Interaction page shows, this way of formulating stock selection means that the value of the stock selection term changes when the allocation of assets changes.  In other words, combining the attributes in this way does not eliminate the interaction term.  Rather, it merely combines the interaction term with the stock selection term.

Another way of dealing with the interaction term is by adding it to the asset allocation term.

In order to achieve an easier presentation of results, there can be good arguments for combining the interaction term with other terms.  However, it is a simple fact that, under the sector attribution paradigm, part of the value added flows from a combined effect of asset allocation and stock selection.  It therefore seems prudent to maintain the ability to disclose the interaction term separately, even if one may sometimes choose to combine it with the other terms in the interest of avoiding the question of how the interaction term works.

Worked Example

Let us now calculate sector-level attributes for the example that appears higher on this page, in the First Principles section.  In that example, we calculated only the total-level attributes.  The sector-level attributes look like this:

For full details of the calculations (and some interesting notes) see the spreadsheet Comparing_BHB_BF.xls.

The totals of these sector-level results do indeed agree with the total-level results that we obtained above.  Moreover, the following commentary shows how each number in the results is susceptible to an intuitive interpretation:

• In asset allocation, equities were 10% underweight in a sector whose benchmark return was 1% below the return for the overall benchmark.  This bet generated 10 basis points of value added.  Bonds added a further 10 basis points of value, by being 10% overweight in a sector whose benchmark return was 1% above the return for the overall benchmark.
• In stock selection, equities made a big contribution, because the portfolio sector return exceeded the benchmark sector return by 2%.  Since the benchmark weight of equities is 50%, this performance added 50 basis points to the portfolio return.  Bonds made a smaller contribution to value added.  The portfolio sector return for bonds exceeded the benchmark sector return by 1%.  Since the benchmark weight for bonds is 50%, the value added attributable to stock selection for bonds is 50 basis points.
• In regard to interaction, the equities sector was 10% underweight, while the sector outperformed its benchmark by 2%.  Thus 20 basis points of performance was lost.  If the asset allocation had been neutral, or if the stock selection had been neutral, this 20 basis point loss through interaction would have been avoided.  Turning to the bonds sector, it was 10% overweight, and it outperformed its benchmark by 1%.  Thus, there was 10% of added value that only arose due to the simultaneous overweight position in a sector that was outperforming its benchmark.

Variations on this Approach: Mistaken or Beneficial?

Now we've established that the First Principles (Above) and Equations 1-3 (Above) provide a consistent and intuitive approach to sector attribution, we can pick up a thread that we left hanging earlier on this page.  Specifically, while deriving the sector-level Equations 1-3, we stopped momentarily to examine a different way of expressing the asset allocation attribute.  The inner term of this weighted sum looked like the right-hand-side of Equation 4:

[ Equation 4 ]

As we noted above, some authors have termed this "the Brinson Hood Beebower" (BHB) method for calculating the value added from asset allocation.  Supposedly, this is because it was described by those authors in their paper published in 1986.  However, the original source does not support this interpretation.  As we noted above, Brinson, Hood, and Beebower were concerned with quantifying how much value was added to total portfolio performance by each source of active performance; and at no stage did they consider sector level performance.  For example, their caption for Table III reads: "Calculation of Active Contributions to Total Performance" (p. 41, italics added for emphasis).  It simply seems to be the case that Brinson, Hood, and Beebower did not address the question of calculating sector-level attributes in their paper.  So, we suggest describing the method for calculating sector-level attributes using Equation 4 as the "so-called BHB method", since it seems historically inaccurate to suggest that the BHB paper suggested anything at all about how to calculate sector-level attributes.

By reviewing the algebra, we can see the likely source of confusion.  Equation 1 provides a different value for the sector-level attribute than Equation 4 does.  However, if one sums these values across all the sectors, the resultant total-level values are the same (one can verify this by reviewing the above derivation of Equation 1 from first principles).  Brinson, Hood, and Beebower only described calculations at the total level, so the evidence seems slender that they meant to advocate a different method for calculating sector-level attributes.

Using the worked example that we considered in the previous section, let us consider whether the so-called "BHB method" of calculating sector-level asset allocation is a useful tool, or merely a trap for the unwary.

The attribution results using the so-called BHB method for calculating asset allocation are:

(Note: Refer to the spreadsheet Comparing_BHB_BF.xls for full details of the calculations and some interesting comments).

As you can see, this seems to make an altogether different estimate of the payoff from each of the asset allocator's two bets:

• For the 10% underweight on equities, our result was that this bet added 10 basis points of value to the portfolio, since the asset allocator underweighted a sector whose benchmark return (2%) was lower than the overall benchmark return (3%).  But the BHB show this underweight position as having cost 20 basis points of performance.  If a portfolio manager were to use the BHB numbers as her guidance, shouldn't she say, "OK, it cost me 20 basis points to be underweight equities, so surely I should do well if I overweight equities as much as possible".  The problem with this is that equities was the asset class with clearly the lowest benchmark return.  Overweighting equities is exactly the worst asset allocation strategy available in this example.
• For the 10% overweight on bonds, it is true that this bet added value to the portfolio, but 40 basis points seems like an excessive estimate of how much value was added.  After all, the bonds benchmark only outperformed the overall index by 100 basis points, so a 10% overweight would seem to lead to a more reasonable figure of 10 basis points of value added.  Taking this principle to extremes, if the returns in the example were all shifted up by 20%, the overall benchmark return would be 23%, and the benchmark return for bonds would be 24%.  So in that case, a 10% overweight to bonds would supposedly add 10% x 24% = 240 basis points to the asset allocation attribute!  This is a big reward for an action that is only going to truly add 10 basis points to the portfolio value.

In summary, the outputs from the so-called BHB asset allocation calculations seem to produce numbers that can be at wild variance to what one expects based on the principles of portfolio management.  We regard the so-called BHB model as a misunderstanding that has taken hold over time.

The following quote from a paper by Steve Campisi touches on some of the confusing issues that surround the idea of that there is a separate BHB attribution model:

"The attribution model introduced by Brinson, Hood, and Beebower was seriously flawed in its calculation of the allocation effect.  The problem with the model was its tendency to attribute excess return to the overweighting of any sector with a positive return, even if that sector reflected underperformance relative to the overall benchmark.  Clearly, overweighting an underperforming sector is a source of underperformance, not outperformance.  The model's error stems from its definition of the allocation effect as the simple difference between the contributions to return for the portfolio and the benchmark sectors."

"A more appropriate calculation of the allocation or timing effect actually appeared in an earlier article by Brinson and Fachler (1985)."

[...] "Since we have clearly shown a significant problem in an accepted attribution model, it is reasonable to check whether any other problems exist." [Campisi, S., op. cit., pp. 65-65]

This text raises several issues:

• It accurately describes the way that Equation 4 provides measure of sector-level value added which is in some cases completely misleading.
• It accurately describes the widely-held belief that there are two separate models for sector attribution (BHB and BF), which respectively use Equation 4 and Equation 1 as their measure of the sector-level value added by asset allocation.
• It reflects the widely-held idea that Equation 4 was proposed by Brinson, Hood and Beebower in their 1986 paper.  The notable thing about this is that, if one examines the text of the paper, it only seems to be about calculating total level value added.  This is where our viewpoint about sector attribution diverges.  It is true that many (most?) people believe that the BHB paper proposed Equation 4 as a different measure of sector-level asset allocation, but we simply cannot see how the text of the paper justifies this belief.  We consider that the widely-held belief that BHB proposed a different attribution model is an incorrect belief.

So, in summary, it seems to be widely believed that there are two separate yet similar attribution models (BHB and BF) for sector attribution.  They differ only in their measurement of the sector-level value added by asset allocation (the total-level value is the same).  The idea that BHB and BF intended to propose different ways of measuring sector-level asset allocation seems just plain wrong because BHB only seemed to address the issue of total-level attribution.  In any event, if BHB is a separate model, hardly anybody seems prepared to argue that it can be justified as a model that makes sense.  So, either way, we suggest that the industry would be better off if it just abandoned the whole idea of a BHB model.

Extending the Model to Include Transaction Cost Measurement

In the Stock Level Attribution page, we developed an approach for measuring transaction costs during stock-level attribution analysis.  A similar method can also be applied to sector attribution.  We explain this approach on the Adding Transaction Cost Measurement page.

Zero-Weighted Sectors

Special cases can arise when the attribution includes sectors whose benchmark or portfolio weight is zero.  On the Zero-Weighted Sectors page, we explain why these are indeed special cases, and how you can deal with them.

Providing Drill-Down to Stock-Level Results

A "drill down" for a sector-level effect (e.g. stock selection) can be done by showing how that effect arises from stock-level effects (these stocks will be the stocks contained in the sector).

Sector Allocation is clearly a decision made at the sector-level (by attempting to tilt the portfolio toward better-performing sectors).  Also, interaction, by its very definition, includes aspects of sector allocation (along with aspects of stock selection).  Hence, Sector Allocation and Interaction are terms in the attribution analysis that are not susceptible to "drill-down".

The page Drill-Down to Security-Level Data in Sector Attribution shows how you can calculate sector attribution with a drill-down to stock-level results (for the stock selection and transaction cost effects).

Moving on to Multiperiod Examples

This page has provided a very useful summary of how to calculate attributes over single measurement periods (e.g. a day or a month).  However, to gain the most insights from attribution analysis, one has to look at the question over a period of months or years.  This requires somehow combining the single-period attribution results in order to obtain a result over multiple periods.  This interesting issue is explored at some length in the papers available on the Multiperiod Attribution page.

References

The following papers are some of the earliest known explanations of how one can determine the value added by asset allocation and stock selection by using weighted sums to create notional portfolios:

Brinson, Gary P., and Nimrod Fachler, “Measuring Non-US Equity Portfolio Performance,” Journal of Portfolio Management, Spring 1985, pp. 73-76.

Brinson, Gary P., L. Randolph Hood, and Gilbert L. Beebower, “Determinants of Portfolio Performance,” Financial Analysts Journal, July-August 1986, pp. 39-44.

Brinson, Gary P., Singer, Brian D., and Gilbert L. Beebower, “Determinants of Portfolio Performance II: An Update,” Financial Analysts Journal, May-June 1991, pp. 40-48.

(Working Group of) The Society of Investment Analysts (UK), “The Measurement of Portfolio Performance for Pension Funds” (1972, Revised 1974).

Notation

The equations on this page use the following notation:

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