I was teaching an attribution class recently, and a student asked for a simple explanation as to why arithmetic attribution effects don’t link (geometric do, which is one of its advantages over arithmetic). The person recognized that the effects don’t link, as do their clients, but he was still wanting to have a pithy response to the question when posed by clients. Simply saying “because they don’t” didn’t seem to work.
Well let’s consider two other items before we answer this question: returns and excess returns. Returns link, right? And why is this? Because returns compound. That is, returns build upon the performance of prior periods. If I start with $10,000 and have a 10% return in January, $1,000 is added to my value and the portfolio ends the month at $11,000. Then if I have another 10% return in February, I don’t add another $1,000, but another $1,100, which is 10% on the initial $10,000 plus 10% on the added value ($1,000) in January; i.e., $1,000 + $100. Returns compound and therefore we link from month to month: arithmetic linking (i.e., simply adding returns together) don’t take the compounding effect into consideration; thus we use geometric linking.
Excess returns (i.e., portfolio return minus benchmark return) don’t link. Why not? Because excess returns themselves don’t compound. And while the portfolio and benchmark returns compound, they may compound in different fashions depending on their individual results. But excess returns don’t compound.
Attribution effects reconcile to excess returns, right? And if excess returns don’t compound how can attribution effects? But we want to be able to reconcile to the linked period excess return, which is based on taking the difference between the linked period returns (what a mouthful!). We accomplish this through a smoothing technique, such as the ones developed by David Cariño, Jose Menchero, and Andrew Frongello (and no, you don’t have to have an “o” at the end of your name to develop such a linking method (but it can’t hurt!) The French group, GRAP, also developed a method to link attribution effects).
To summarize, attribution reconciles to excess returns. Unlike the returns themselves, arithmetically derived excess returns don’t compound. Therefore arithmetic attribution effects don’t compound. In the case of geometric attribution, their excess returns do compound, so the attribution effects compound, too.
Hopefully this makes sense, though I’m open to your thoughts.