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.
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You are doing something quite useful in seeking an understanding of what is actually causing these performance results, rather than simply getting caught up in the measurement procedures. Your explanation of linking in money terms makes this tangible by relating the beginning amount to the ending amount. This is by definition a geometric approach, rather than an arithmetic approach, which has the very nice quality of making a series of individual periods equal to a longer single period. Investors essentially want a return that reconciles their beginning capital to their ending capital, something that requires a geometric approach and linked returns.
ReplyDeleteIt is also critical to understand that this analysis is based on the assumption of a single amount that remains invested without additions or withdrawals of capital. If capital is added or withdrawn, then even the geometric approach will not suffice for calculating return or excess return. In that case, the only method that works is a money weighted approach, and it works for the simple reasons that a) it creates a single period for the analysis and b) it reconciles the beginning capital and change in capital with the ending capital.
We often ignore or forget these assumptions, and this causes the lack of "intuitiveness" that you discuss. The simple question that we must always remember is to ask: "What are we measuring?"
Steve, as always I welcome and appreciate your insightfulness. Your last comment is a very important one: what are we measuring? This should ALWAYS be a key question that is considered. Too often we simply put numbers out there without thinking of what they represent and the value they have for the recipient. And, giving someone loads and loads of numbers and statistics that we think demonstrate how smart we are and how extensive our system is does nothing if those numbers don't represent something of value to the client. Many in the industry are coming to this realization, which is a great thing.
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