It’s probably appropriate to begin by explaining what attribution is. In general terms, it’s essentially a process to assign blame or credit for some event occurring. In the world of performance measurement, it has to do with identifying responsibility for our ability to beat (or failure to beat) the benchmark. We want to know the cause(s) for our success (or failure), thus the interest in attribution.
Attribution linking is gaining attention for various reasons; one is the movement to daily attribution. No one actually wants to look at the daily attribution results; rather we use the daily numbers to achieve greater accuracy in our monthly (or quarterly or longer) attribution figures.
The problem with the monthly models is that they don’t account for intra-month activity. Unless our portfolio remains pretty static for the period, we’ll encounter some error when we use a monthly process. Consequently, more and more firms are moving to daily attribution models. They use the daily numbers to improve the precision of their monthly (and quarterly and annual) reported attribution results.
Well, how do we get the monthly effects from the daily? By linking. And there’s the rub. It’s not easy. It’s fairly simple with geometric models of attribution; the challenge occurs when we use an arithmetic (or additive) model.#
The primary difference between geometric and arithmetic models is how they look at excess return (also called active return). In the arithmetic model, it’s simply the difference between the portfolio and benchmark returns (portfolio minus benchmark). With geometric models, it’s presented in fractional format (portfolio return over benchmark return). Because of the natural way geometric links we don’t have the same problem with these models as we do with arithmetic. Therefore, this article deals with the challenges of linking results when using arithmetic models.
In this article, I’ll summarise some of the key aspects surrounding this increasingly important topic, without providing the detailed maths behind the approaches. Those interested in the maths are invited to read either my forthcoming book or the articles I’ve cited.
In the book, I promulgate some ‘laws’ for attribution. For example, the first law essentially states that the attribution model we use should conform to our investment approach. The second law is ‘the sum of the attribution effects must equal the excess return’. I don’t believe this law will be challenged, as it is basic to the concept of attribution, where we attempt to discern the causes of the excess return (also referred to as ‘active return’ or ‘alpha’).
The second law states that the sum of our attribution effects must equal our excess return. And, the third law builds upon the second: ‘the sum of the linked attribution effect must equal the sum of the linked excess return’.
What’s an attribution effect? These are those factors that we’ve decided to investigate as potential causes for our result. These ‘effects’ take various forms, depending on our style of investing. For example, with equities, we may be looking at our stock selection decisions, our asset allocation activities, or our decisions to over or under-weight countries, relative to the benchmark. A key is the notion of ‘relative to the benchmark’, as we are exploring the manager’s decisions relative to the benchmark. For fixed income managers, we may look at the effect of duration, spread, or other analytical events commonly analysed by bond managers.
Whether we’re linking daily numbers to get to monthly, or monthly numbers to arrive at quarterly or annual, the same major issue arises: how to do it.
We’ll discuss four approaches to linking attribution effects: two rather simple and two more challenging. At the end, we’ll discuss how most vendors seem to approach this challenging topic.
At first glance, we’re tempted to geometrically link the attribution effects. After all, isn’t that the way we arrived at our linked returns? But, it isn’t that simple, because when you do the maths, you usually end up with unaccounted for and unexplained residue. Since we like the numbers to add up properly and since there’s no assurance with geometric linking that they will, we have to pass on this approach.
What about arithmetic linking? Well, our numbers work out in a way that the results are identical to geometric linking, so this doesn’t work, either.
Since these basic and simple approaches fail, we have no other choice than to move to more complex and mathematically rigorous approaches. One was developed by the Frank Russell Company and used in its attribution system.# The maths is far from trivial, and interested readers are invited to explore David Cariño’s article (‘Combining Attribution Effects Over Time,’ The Journal of Performance Measurement, Summer 1999), which details the process.
Cariño’s method seeks to find ‘k-factors’ which are used to adjust the attribution effects to satisfy our third law of attribution. These factors are what he develops to ensure that the results tie out with our excess return.
A similar approach was developed by José Menchero of Vestek and called the ‘optimised method’. Like Cariño’s approach, this uses some fairly non-trivial formulas (‘An Optimized Approach to Linking Attribution Effects Over Time,’ The Journal of Performance Measurement, Fall 2000).
Both Cariño and Menchero are PhDs. I believe the additional ‘brain-power’ that they obviously possess is necessary to develop the sophisticated models necessary to overcome the shortcomings of arithmetic or geometric linking.
Our preliminary research has shown that most of the other performance attribution software vendors use geometric linking (see The Journal of Performance Measurement Technology Supplement, 2002).# In some cases, they don’t worry about the fact that they violate our third law. In other cases, they use what might be called a ‘smoothing’ technique to get the numbers to add. Often, these techniques identify the difference between the calculated results and what was expected, and then allocate the difference proportionately across the effects.
Some vendors are planning to use either the Cariño or the Mencheromodel. We believe that a few are attempting to develop their own, mathematically robust models.
It’s also interesting to note that some people disagree totally with the notion that our numbers are supposed to add up. The general premise seems to be that the notion that attribution effects are supposed to compound (as our returns do) is a fallacy.
As noted earlier, geometric attribution isn’t challenged in the way arithmetic attribution is. This is one of several reasons why many firms are considering the geometric approach.
While performance attribution has been around for 20 years, it’s still undergoing significant change, as we explore better ways to do the analysis and to address the shortcomings we identify.
When planning to implement an attribution system, there are a number of things to consider; one of the increasingly important ones is how to link results over time. I hope that this article has given you some insight into some of the approaches and reasons why this is an important step in the process.
David Spaulding is president of the Spaulding Group, based in Somerset NJ. His book, Investment Performance Attribution, will be published by McGraw-Hill in the autumn

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