Attribution Basics

This page introduces the fundamental principles of performance attribution calculations.

Before moving on to attribution, we start with contribution, which is a similar (but simpler) concept. Contribution calculations are an important building block in attribution calculations.

Contribution Calculations

A contribution calculation is perhaps the most intuitive of all performance calculations. For example, suppose that your personal investment portfolio is in three separate products (all equity-based) offered by different fund managers: a large-cap fund, a mid-cap fund, and a small-cap fund. When receive your monthly statements from these fund managers, you will most likely want to calculate the overall return achieved by all your holdings. Naturally, you'll also want to be able to judge what portion of your overall return came from each of the three managed funds that you hold.

This example illustrates how to do a contribution calculation over a single period (e.g. a month).

Suppose that the market value of each different fund for the period we are considering were as follows:

Sector	Start Value	End Value
Large Cap	$20.00	$22.00
Mid Cap	$50.00	$59.00
Small Cap	$30.00	$30.60

We can use this information to calculate the (opening) weight and return for each sector, as shown in the following table:

Sector	Weight	Return
Large Cap	20%	10%
Mid Cap	50%	18%
Small Cap	30%	2%

In each case, the opening weight of a sector is simply its start value divided by the sum of the start values.

The percentage return for each sector is simply [(EndValue / StartValue) - 1] X 100.

Now, we get to the interesting bit. Using the sector weights and returns, it is possible to calculate the contribution of each sector to the total return. The contribution is simply the weight multiplied by the return:

Sector	Weight	Return	Contribution
Large Cap	20%	10%	2.0%
Mid Cap	50%	18%	9.0%
Small Cap	30%	2%	0.6%

Total

100%

11.6%

What this tells us is that the total return of the portfolio was 11.6% (or, in other words, 1160 basis points). Of these 1160 basis points, 900 were contributed by the Mid Cap sector. 60 basis points were contributed by the Small Cap sector. And the Large Cap sector contributed exactly 200 basis points.

Note that the total return should (by definition) have the same value as the sum of the sector contributions. If this fails to reconcile, it most likely indicates that there was a minor calculation error somewhere along the way. Like a lot of performance calculations, one can check this calculation for mistakes by doing a reconciliation. The calculation should reconcile according to the precision with which the computer can do arithmetic.

Contribution is a very simple concept, but it plays an important role in various kinds of investment performance analysis.

Weighted Sums

The contribution calculations take us on to another vital area: weighted sums. A great portion of all the calculations done by an investment performance analyst will take the form of weighted sums.

What is a weighted sum? We saw an example in the preceding paragraph. To obtain the total contribution of all the sectors in the portfolio, we did the equation

contribution = (20% X 10% ) + (50% X 18%) + (30% X 2%) = 11.6%.

In Microsoft® Excel™, the normal way to implement such a calculation is with the SUMPRODUCT function that is supplied with Excel.

When doing algebra on paper, it would become exceptionally tiresome to write-out in detail all the numbers in a weighted sum calculation. Furthermore, it's useful to use algebra to solve a particular kind of problem independently of the specific inputs that will be supplied to this problem from time to time. Therefore, in algebra one normally uses "Sigma Notation", where the Greek letter Σ (Sigma) indicates a summing operation. For example, we might rewrite the above contribution calculation as:

contribution = Σ w.r

In other words, the contribution calculation is done by multiplying each sector's weight with its return, and then summing the results.

Various kinds of weighted sum calculation appear all over the place in portfolio analysis problems. Simply by using different vectors of returns, or different weight vectors, one can express ideas about what a portfolio's performance would have been, had certain assumptions applied.

Notional Portfolios

The concept of a notional portfolio is critical to understanding performance attribution. In this context, a notional portfolio consists of combining a set of sector weights and sector returns. Everybody is familiar with one particular kind of notional portfolio: a benchmark. For example, for the above portfolio, the benchmark specification would contain the following information for each sector:

An index that can be used for measuring the performance that the sector might obtain if managed passively; and
A weight for each sector. This weight is considered to represent the "neutral" asset allocation of the fund.

In a benchmark that has been specified properly, the benchmark weights sum to 100%.

So one kind of notional portfolio can be obtained using the benchmark sector weights and benchmark sector returns. This notional portfolio has returns, which can be compared with the portfolio returns.

Other notional portfolios can be obtained by combining:

the benchmark sector weights with the portfolio sector returns; or
the portfolio sector weights with the benchmark sector returns.

Both of these notional portfolios are used in performance attribution, to help measure the value added from asset allocation and stock selection (respectively). For more detail, see the Sector Attribution page. However, at this stage, the key point to grasp is that a notional portfolio is simply a set of sector weights and sector returns, which produces a total-level return that one can use for the sake of comparison with the portfolio return. In its essence, performance attribution is all about comparing notional portfolio returns.

Three Building Blocks of Performance Attribution

From a mathematical standpoint, the essence of performance attribution comprises three very simple operations:

Weighted sums;
Compounding; and
Value-Added.

Weighted sums are explained above. Compounding is of course an elementary performance calculation (See the Measurement Basics page). Value-added is demonstrated thus: when a portfolio outperforms its benchmark by x%, there has been x% of value-added.

Together, these three principles are important building-blocks for performance attribution. Problems such as multiperiod attribution or multicurrency attribution are not so difficult when you retain in your mind a clear understanding of these three principles.

Two Key Considerations for Treating any Instrument Sensibly in Attribution Analysis

Questions frequently arise about how to deal with derivatives or unusual instruments. There are two considerations that will go a long way toward clarifying how the instrument should be treated.

1. What is the weight of the instrument in the portfolio? For many instruments, determining the portfolio weight seems rather easy. However, for derivatives, it is not entirely obvious. The main thing to consider is the exposure of each holding. See the Measurement Basics page for information about exposure and portfolio weights.

2. What is the return of the instrument? Once again, leveraged instruments are the source of most difficulties. Another source of confusion is short positions (should the sign of the return be reversed for short positions?) A sound grounding in performance measurement will help you with this sort of problem in performance attribution.

At this stage, it isn't really practical to explain the right way to treat these two considerations for every possible instrument. However, whenever you have problems about how to treat an instrument in performance attribution, the above two questions will help you to move in the general direction of a good solution.

These two considerations solve a lot of the mysteries concerning different instruments. However, they don't solve everything. If only life were that easy!

Return Attribution vs. Risk Attribution

The broad approach to performance attribution in this web site has sometimes been described as "return attribution", in contrast with "risk attribution". Another term that has been used to describe this web site's approach to attribution is "accounting-based attribution". That is because the attribution techniques presented on this web site exactly decompose the active return into several components, using a calculation method that is susceptible to reconciliation.

The "risk attribution" approach is based on factor models. Factor models are a statistical approach to explaining portfolio risk.

To put it in a nutshell, return attribution explains portfolio performance in terms of overweighting particular stocks, sectors, or currencies. Risk attribution takes a more statistical approach, explaining portfolio performance in terms of overweighting factors in a statistical model of investment performance.

Choosing between these approaches is not a matter of "right or wrong". It is more a matter of deciding which approach will give more insight to the particular audience that you are targeting.