Statistical Interactions and Performance Attribution

Tuesday, May 29, 2007 9:00 AM

About two months ago, I attended a talk at the New York Society of Security Analysts (NYSSA) on performance attribution for investment portfolios. Although there was some discussion of attribution of fixed income portfolios, most of discussion was focused on equity portfolios and two Brinson models. The Brinson models basically divide performance attribution into two dimensions: returns attributable to overall allocation decisions, and returns attributable to security selection decisions. The interesting wrinkle to this analysis is that these effects interact with each other, and as a former quantitative instructor, I thought it was interesting that a room full of experienced analysts was still having trouble understanding interactions.

This commentary is about interactions and how to understand them, but first, I need to fill in a little background on performance attribution.

For the Brinson models and equity portfolios, allocation and security selection are less self evident than they would seem. Asset allocation is the attempt to improve investment returns or control risk by selecting which groups of securities are likely to perform well and how they behave relative to each other. Typically, these groups of securities are defined by industry (e.g. transportation, financials, consumer durables, etc.), but they could equally be defined by countries (e.g. US, Japan, China, etc.), or even something less traditional, such as segments of a global production chain or based fundamental factors like market capitalization. Once one has defined the categories that define “allocation,” the Brinson models essentially divide a portfolio’s performance into the portion of returns that can be attributed to decisions “between categories” (allocation), and the portion of returns that can be attributed to decisions “within categories” (security selection). An interesting implication is that the exact same portfolio could have different amounts of attribution to security selection, depending on how one has defined the categories. For this reason, incidentally, I have always found the Brinson techniques intellectually interesting, but ultimately unsatisfying, because the categories seem more or less arbitrary, and may not adequately capture where an investor’s true talent lies.

The interesting thing is that the total performance of a portfolio isn’t simply the sum of the allocation effect and the security selection effect, it is the sum of these two effects plus their interaction. It is actually possible to outperform a benchmark portfolio in every category consistently, but if the wrong categories end up having too much weight, the portfolio underperforms the benchmark as a whole, because of the interaction. Mathematically, the interaction effect appears as a portion of allocation return multiplied by a portion of selection return, and while most of the NYSSA audience had little trouble understanding the mathematics behind the interaction, there was a surprising degree of difficulty grasping the intuition of what was happening.

Sometimes the easiest way to describe mathematical interaction is by comparison with drug interactions. For example, having some wine with dinner might be a nice way to relax in the evening. Taking an antihistamine might be a sensible way to control allergies, which could also affect your ability to relax in the evening. But taking an antihistamine AND having a bottle of wine with dinner could create an entirely new effect - say, convulsions or a heart attack. Most certainly not relaxing. The interaction is something that neither drug induces on its own, but happens when both are taken together. Another way to think about interaction is a synergy, or, in this case, a dys-synergy. These examples suggest that the total effect of one component depends on the total effect of the other.

[Disclaimer: this example is hypothetical for the point of illustration, please read your antihistamine indications carefully; most will advise against taking with alcohol, but I do not know what specific interaction the drug companies are worried about.]


There are essentially two ways to interpret interactions in a causal framework: 1) understanding interactions as forms of synergy; and 2) understanding interactions as moderated effects. These are mathematically equivalent formulations, for the most part, but the way an interaction is interpreted can go a long way to making it understandable.

Suppose you have a model with an interaction term (term Dxy in this example).

R = A + Bx + Cy + Dxy

This formulation is consistent with the idea of synergy or dys-synergy, depending on wether term D is positive or negative. If D is positive, then x and y together have a greater influence on R than either would in isolation (synergy), and if d is negative, then x and y together have a lesser inlfuence on R than one would guess from each of their individual effects (dys-synergy, or mutual interference). This interpretation assumes that coefficient B and C are positive; the story would need to be more carefully constructed if one or both signs were negative.

Now, for a different interpretation, one could factor Y out of the terms Cy + Dxy, and get

R = A + Bx + (C + Dx)y

Which can be interpreted in the following way: R is a function of x and y, but the effect of y depends on the value of x. In a causal framework, where you suggest that x and y actually cause R, you can say that “the effect of y is moderated by the value of x,” or in other words, the degree to which y influences R is a function of the value of x.

Of course, one could equally have factored out x from term B and D of the equation and decided that the influence of x is a function of y. This is a choice up to the analyst and depends on which method of factoring makes the most intuitive sense. In practice, the easiest way is to think clearly about which variable is having its effect moderated by the other, and then factor out the variable that is being moderated.


So what do interactions mean in a situation like Brinson’s allocation models? To take the synergy approach, it suggests that if you are both a good asset allocator and a good security selector, you may achieve even greater returns than your individual skill on each dimension might suggest. Therefore, it is important to have an investment process that allows you to capture that greater return. How much greater depends on a number of things, including investment policy, constraints, etc., but as a rule of thumb, other things equal, it is probably a better strategy to try to improve the weaker dimension than to continue to improve the stronger. This is a conclusion that runs counter to most managerial instincts, particularly in an industry focused on differentiation and maintaining a unique edge. However, there is no reason that managerial techniques cannot organize specific edges into a strong overall process that includes more mundane but necessary decisions. These structures can retain the value of having one or more unique edges, but the process must consciously integrate the interacting factors, and possibly despite inter-firm tensions or rivalries.

Taking a moderated effects approach, the interactions in the Brinson models suggest that the degree to which security selection skill affects returns depends (at least in part) on one’s allocation skill, and one’s allocation skill depends in part on one’s security selection skill. Therefore the managerial process by which one merges selection skill with allocation skill is extremely influential in determining the overall portfolio result. This is essentially the same conclusion as they synergy formulation, but it is often easier to grasp and can often lead to deeper insights into one’s underlying model.